examples/001_Getting_Started.ipynb

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   "source": [
    "import numpy as np\n",
    "# import matplotlib; matplotlib.use(\"SVG\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Getting started with `josiepy`\n",
    "## A Python framework to simulate Partial Differential Equations focused on usability\n",
    "### (without sacrifying performances)\n",
    "\n",
    "<i class=\"fa fa-user\"></i> Ruben Di Battista <br /> \n",
    "<i class=\"fa fa-twitter\"></i> <a href=\"https://twitter.com/rdbisme\">@rdbisme</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
    }
   },
   "source": [
    "## Background\n",
    "----------------------\n",
    "### M. Massot Research Team\n",
    "####  Mathematical modeling\n",
    "- Mathematical modeling of two-phase injection in rocket engines\n",
    "- Mathematical modeling of plasma flows for heliophysics and hall thrusters\n",
    "\n",
    "#### Numerical Calculus for High Performance Computing (HPC)\n",
    "- Development of numerical schemes and algorithms adapted to HPC\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## `canoP`\n",
    "### A C++ wrapper on `p4est`, a quad/octo-trees for parallel AMR computation, to simulate PDEs w/ AMR\n",
    "\n",
    "- C++ is notoriously hard to master\n",
    "- `p4est` is an advanced library w/ features not immediately needed when exploring numerical schemes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Motivations\n",
    "--------------------\n",
    "\n",
    "An easy to use playground to easily implement numerical schemes on 1D, 2D (and maybe 3D in the future) structured (and maybe unstructured and non-conformal meshes... dreaming is still free 😇) meshes, in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Design Choices\n",
    "--------------------------\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Use Python instead of a Domain Specific Language (DSL), like the one OpenFOAM uses, or another scripting language, like `lua` in `canoP` to describe your case configuration. \"You code your case configuration\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Modern Python (Python >= 3.7) features allows static checking (through the usage of type hinting) of the code that predates some advantages of the compiled languages"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- The `numpy` library and its API allows to write algorithms once while being able to run them on different architectures and different programming paradigms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "   - CPU Shared memory ([numpy](https://numpy.org) + OpenMP-enabled BLAS library)\n",
    "   - CPU distributed memory ([Dask](https://dask.org))\n",
    "   - Nvidia GPUs with CUDA ([cupy](https://cupy.chainer.org))\n",
    "   - (Soon) AMD GPUs with HIP ([cupy](https://cupy.chainer.org))\n",
    "   - ... FPGA? Anyone? It's a cool project too... \n",
    "   \n",
    "   Everything mostly at the same computational speed as C and C++, or faster (GPU for example)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "\n",
    "## Numpy for `josiepy` 101\n",
    "### Vectorized operations\n",
    "\n",
    "What you would generally do in a C++-like compiled code is: \n",
    "\n",
    "```\n",
    "for( cell : mesh.cells() ) {\n",
    "    apply_compute_kernel(cell)\n",
    "}\n",
    "\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "What we generally will do in `numpy` is operating on an array containing **all** the cells values at the same time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's imagine we have the velocity vector stored in an array of size $N_\\text{cells} \\times \\text{dim}$, where $\\text{dim}$ is the dimensionality of the problem (e.g. 2D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "cell_values = np.random.random((30, 2))\n",
    "U = cell_values[:, 0]\n",
    "V = cell_values[:, 1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let's imagine now to have to compute a normal product between all the velocity values in each cell, and a singular \n",
    "normal vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "normal = np.array([1, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "What we could do is repeating the `normal` array $N_\\text{cells}$ times and multipliying it by `cell_values`..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "... but there's a better way!\n",
    "The operation we want to do is\n",
    "\n",
    "$$\n",
    "    \\left\\{\\mathbf{U}\\right\\} \\cdot \\mathbf{n} = U_{ij}n_j\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(30,)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "normal_velocities = np.einsum(\"ij,j->i\", cell_values, normal)\n",
    "print(normal_velocities.shape)\n",
    "\n",
    "np.array_equal(U, normal_velocities)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "`numpy.einsum` is an implementation of the Einstein's tensor summation on steroids 💪. You specify the input vectors dimensions and the output dimension you want to achieve, and it figures it out automatically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "Basically:\n",
    "$$\n",
    "U_{ij} n_j \\rightarrow u_i\n",
    "$$\n",
    "```\n",
    "np.einsum(\"ij,j->i\", cell_values, normal)\n",
    "```\n",
    "* Repeated indices on the inputs are summed as a result if they're not present in the result (after the `->`): `cell_values[i][0] * normal[0] + cell_values[i][1] * normal[1]` "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "`np.einsum` avoids memory copies and if we provide the `optimize=True` keyword argument"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.08 ms ± 20.3 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "A = np.random.random((1000, 1000))\n",
    "B = np.random.random((1000, 1000))\n",
    "\n",
    "%timeit np.einsum(\"ij,ij->\", A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "684 µs ± 158 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "A = np.random.random((1000, 1000))\n",
    "B = np.random.random((1000, 1000))\n",
    "\n",
    "%timeit np.einsum(\"ij,ij->\", A, B, optimize=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "it is gonna exploit parallelization of operations like `np.dot` and `np.tensordot` that are parallelized natively with OpenMP if the BLAS library which `numpy` is compiled on allows it \n",
    "\n",
    "(hint: the classical `pip install numpy` pre-compiled wheel, does not)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Vectorized conditionals\n",
    "\n",
    "Sometimes, for example to initialize the domain at the beginning of a simulation, or for an upwind scheme, you need to apply some conditionals of the type:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "```\n",
    "for (cell : mesh.cells()) {\n",
    "    x, y = cell.centroid()\n",
    "    \n",
    "    if(x > 0.5) \n",
    "        right_state(cell)\n",
    "    else\n",
    "        left_state(cell)\n",
    "}\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "What you would do in vectorized form, is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0.]\n"
     ]
    }
   ],
   "source": [
    "values = np.zeros(30)\n",
    "x = np.linspace(0, 1, np.size(values))\n",
    "print(values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 1. 1. 1. 1. 1. 1. 1. 1.\n",
      " 1. 1. 1. 1. 1. 1.]\n"
     ]
    }
   ],
   "source": [
    "values[np.where(x > 0.5)] = 1\n",
    "\n",
    "print(values)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Basics of `josiepy`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "### Problem statement\n",
    "\n",
    "\\begin{equation}\n",
    "    \\partial_t \\mathbf{q} + \n",
    "    \\nabla \\cdot \\left(\\underline{\\underline{\\mathbf{F}}}(\\mathbf{q}) \n",
    "        + \\underline{\\underline{\\underline{\\underline{D(\\mathbf{q})}}}} \\cdot \\nabla \\mathbf{q} \\right) +\n",
    "    \\underline{\\underline{\\underline{\\mathbf{B}(\\mathbf{q})}}} \\cdot \\nabla\\mathbf{q} = \n",
    "    \\mathbf{s}(\\mathbf{q})\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "    \\partial_t q_p + \n",
    "    \\partial_{x_r} \\left(F_{pr}(q_p) + D_{pqrs}(q_p)\\partial_{x_s}q_q\\right) + \n",
    "    B_{pqr}\\partial_{x_r}q_q = \n",
    "    s_p(q_p)\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The `State` object\n",
    "\n",
    "\\begin{equation}\n",
    "    \\partial_t \\mathbf{q} + \n",
    "    \\nabla \\cdot \\left(\\underline{\\underline{\\mathbf{F}}}(\\mathbf{q}) \n",
    "        + \\underline{\\underline{\\underline{\\underline{D(\\mathbf{q})}}}} \\cdot \\nabla \\mathbf{q} \\right) +\n",
    "    \\underline{\\underline{\\underline{\\mathbf{B}(\\mathbf{q})}}} \\cdot \\nabla\\mathbf{q} = \n",
    "    \\mathbf{s}(\\mathbf{q})\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "Represents the $\\mathbf{q}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.state import State\n",
    "from josie.fluid.fields import FluidFields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "class EulerFields(FluidFields):\n",
    "    rho = 0\n",
    "    rhoU = 1\n",
    "    rhoV = 2\n",
    "    rhoE = 3\n",
    "    U = 4\n",
    "    V = 5\n",
    "    rhoe = 6\n",
    "    p = 7\n",
    "    c = 8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.21800124 0.44334009 0.31521554 0.33580545 0.28092686 0.77100948\n",
      " 0.0988216  0.71725491 0.21517077]\n"
     ]
    }
   ],
   "source": [
    "class EulerState(State):\n",
    "    fields = EulerFields\n",
    "    \n",
    "rnd_state = np.random.random(len(EulerFields)).view(EulerState)\n",
    "print(rnd_state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "U1 = rnd_state[EulerFields.U]\n",
    "U2 = rnd_state[4]\n",
    "U3 = rnd_state[-5]\n",
    "U4 = rnd_state[rnd_state.fields.U]\n",
    "\n",
    "assert np.array_equal(U1, U2) and np.array_equal(U2, U3) and np.array_equal(U3, U4)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "`State` can be multidimensional! (e.g. a 2D 100x100 mesh, each cell containing an Euler state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "rnd_state = np.random.random((100, 100, len(EulerFields)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100, 100)\n"
     ]
    }
   ],
   "source": [
    "U = rnd_state[..., EulerFields.U]\n",
    "print(U.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The `Ellipsis` object (`...`) allows to index on the last axis, whatever dimension the array is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "rnd_state = np.random.random((100, 100, 100, len(EulerFields)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100, 100, 100)\n"
     ]
    }
   ],
   "source": [
    "U = rnd_state[..., EulerFields.U]\n",
    "print(U.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let's compute the normal velocity again..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100, 100, 100, 2)\n"
     ]
    }
   ],
   "source": [
    "UV = rnd_state[..., EulerFields.U : EulerFields.V + 1]\n",
    "\n",
    "print(UV.shape)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100, 100, 100)\n"
     ]
    }
   ],
   "source": [
    "U_norm = np.einsum(\"...ij,j->...i\", UV, normal)\n",
    "print(U_norm.shape)\n",
    "\n",
    "assert np.array_equal(U_norm, U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The `Mesh` object"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "A `mesh` object is used to create an manage a mesh. A `mesh` is built from `BoundaryCurves`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.boundary import Line, CircleArc\n",
    "\n",
    "left = Line([0, 0], [0, 1])\n",
    "bottom = CircleArc([0, 0], [1, 0], [0.5, 0.5])\n",
    "right = Line([1, 0], [1, 1])\n",
    "top = Line([0, 1], [1, 1])\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DNWuyf214YFwDqzlpPvQQzYeep3SksnUt4KnL6C+D74+Efz81+9dl1nyr6qR3yPVe9xmYWhfF4c6T4IGLgmsQwj9z869Qp56hFPOhJyZyUYaep6hIpb0FdrwPn50dXGi4cIYGM2W3PRlcrekHf/k5ePc1WP5A569RJNOJMvQQZeh52PxG9kilrQXGfTl4XNdfg5mSv1yDq9cNJnJKgqzXKvROqkMPUR16SlT52IAh8PRcePW/s39dZqSiDlwK0dXPTLaBVRzu+wc4/mI4+CR46f5efSKhOvQQ1aETXXL44FTAYcBQGH819B8Ki67NHamIFEtkJNMXDvkUvPEEvNwIA4fD9k3BX4rQK0sfS1GHntegqJmdaWarzGy1mc3oYru/NTM3s8h5BrpDGTrR+TgeXPBz5Qo47VtwwsUa0JTSihpYnXg7fGkeXPUyTJoTTBSW7szT0jl7LxGLDN3MqoE5wBnAOuBZM2t095UZ2w0EpgFPF611Ib0+Q9+6Lns+vmNzcJOHNEUqUmrZfuZq+wWlsL+8PPrrelHOHpcM/ThgtbuvATCze4FJwMqM7W4AbgKmF611Ib0qQw/n5AOHwb6HwdouJlvKvOBHJG66ytkfuRo+8RfBhGEVnK+XIkPPJ3I5AAi/E+tSyz5mZmOBA939ka5eyMymmlmTmTVt2lRYVNJr6tAz68g/3ABvPgEHnwxn3qgLfiSZoi5gqukb/Fw/+xP47xkVX8deijr0bl9YZGZVwA+Ab+Ta1t3nunuDuzcMHVpYR9xrMvTfXhs9Mdb7q+GES5WPSzJly9n/8ZGOtwZMq8B8PRYZOrAeCF8qVp9aljYQGAMssWBmtmFAo5lNdPf872CRQ8Vn6Ls+hKdugw8j5lCB3Vmj8nFJqmw/ux9myZC3rg1uxF3Xv2fbVSJxydCfBQ43s0MIOvLzgS+mV7r7VmBI+rmZLQGuLmZnDhWWoXeoJT8gKO967bewbWPwZ2nUGbpycqlUWfN14PZxMOHbwf1nH/1OojP2WGTo7t4KXA4sBF4G5rn7CjObZWYTi9aSHComQ+8018o6ePH/Brdt+6dHgz9LlZNLb5JtgrDx04OigIcvhYcuSXzGHpu5XNx9AbAgY1lkD+Pup3S/WZ1VTIYeWUsOtO0M7sNZP273dgk+GxHJW1dTC5zyL3DLYbDjvY5fk8C5YuKSocdCRWTobS3Z/7QM1+MqJ5feJtvPfFVVMLFclITVsMclQ4+FxGXomXOuHHMBvDw/+/bKyEWiZcvYq2tgwzLY9Eoi/qKNRYYeF4nK0KPmJF/yPdiyDk64TBm5SCGiMvbqOqjuC3eNDzL2BOTriahDL5VEZejZcvI+A+DM76mWXKQQUTXsk+bAlS9BbX9ob+24fUxr2JWhhyQqQ8+W7aXv06mMXKQw2T4zLdujt49hvq57ioYk5p6i77wEVdXR65STixRXts/UXhFXn5aZ7ikaEtsMPXxz3BsPhrs+FfwZWN2n43bKyUWKLypfB9j2XnAv3RjduFoZekgsM/TMwc+dW4L/T78eJt2hnFykp0Xl62f/Kww5LLhgL0aDpcrQQ2KZoUcNfno7PPkDuHK5OnCRUojK15/6YeftynwxkjL0kNhl6O75XSQkIqW3dX2W5eX7bCpDD4lVht7aDPOnZV+vwU+R8sr2GawbAG2t0et6WGzmcomDsmfo4Ss/q2uhrRmOOAteX6IbMovETdSNq6tqoPkj+NHJsOuDoIy4hFeWliJDT8wZelkz9MzBz7bm4Eq1MZ/XRUIicRQ1WPq5O2HshbDpZfhgPaUeLNVcLiFlncslavCzrTlYrsFPkXiKGiyNuoK0RIOlmsslpKwZugY/RSpDts9sCT7LqkMPKVuG/vTc7Os0+CmSLNk+s/0G9/iuVYceUrIMPTz42WdgMHiy/9GwaRW0avBTJNGiBkutCnZshv/3T/DW0h6bhld16CElqUPPHPzc9QFYNRx/KUzU4KdI4mWbuXH4WHjp/h69srQUdeiJOUM/deRQ/mvpW5HLiybyys82WPwdDX6KVIqowdLF3+28XZEHS8fXj+e+VfdFLi+WxJyhlyRDL+OAiYiUUQmuLFUdekhJMvQ+A6OXa/BTpLJl+4wX8bOvDD2kRzL0zKlv05l5mAY/RSpftml4B48o2hS8msslZPpnjqS22josq602pn/myD17waipb606uJJMg58ivUunwdJ6GDgc3nyiaAOl08ZOo7aqtsOy2qpapo3tYl6oAiVmUBQAz/G8ENkGQFf/NhgAFZHeJXOw9AejO2/TzYFSd+/yeXcl5gz9loWraGnvePAt7c4tC1ft2QtqAFREupK+B3CmPewjZj8/m1bvONNjq7cy+/nZe/R6URLToRd9UHTv/aOXawBURKDoA6UaFA0p6qBoWwvURVS0aABURNKiBkqtCk779h69nAZFQ7o9OVe4ouWmEfDuKmj4XxoAFZFomQOl/fYJbjG58pd7VPmiG1yEdOvConRFS3oQtPmjYLL7g/4Gzr21iK0UkYoSHih1h5+dA6se2b0+XfmS3rYLurAopFsZelRFS3tr9NzIIiJRzGDLm52XpytfcohNhm5mZ5rZKjNbbWYzItZfZWYrzWyZmS0ys4OL1sKUbmXoqmgRkWLoxhQBscjQzawamAOcBYwGpphZZoHmC0CDux8FPADcXLQWpnQrQy/BZb0i0gt0oy+Jyw0ujgNWu/sad28G7gUmhTdw98Xuvj31dClQ9J6yWxn6X5/feZkqWkSkUFGVLzV98upL4nKDiwOA8D3Y1gHHd7H9RcCvo1aY2VRgKsBBBx2UZxMDBWfo4RtVVFVB7V7Qb1DJ7/QtIhUk3Wek+xYzqO4Hv7seHpzaZd+SuJtEm9kFQAPwqaj17j4XmAvQ0NBQ0DWvBd0kOrOqpb0NrAVOv06duIh0T7jy5Tcz4fezYdeW4HkXVS9xuUn0euDA0PP61LIOzOx04JvARHffVZzm7VZQhh5V1dK2S1UtIlJcKx7svCxL1UtcMvRngcPN7BAzqwPOBxrDG5jZMcBdBJ35xqK1LqSgDF1VLSJSCgX0NbGoQ3f3VuByYCHwMjDP3VeY2Swzm5ja7BZgL+B+M3vRzBqzvNweKyhDV1WLiJRCAX1NbOrQ3X2Bux/h7n/h7t9NLZvp7o2px6e7+yfd/ejUv4ldv2LhCqpDHz+98zJVtYhIsUVWvUT3NbGoQ4+LvDL09Hwt6UGJPnujeVpEpMdkzveS9uDUTvO8aC6XkJwZemZlC0B7C3x+rjpyEek56aqXZfPgoUugNdUHZVS8xCJDj4ucGXpUZUuecyyIiHTbolnBXc/CQn1QbDL0OMiZoauyRUTKKUcfpAw9JGeGrsoWESmnHH1QXOrQYyFnhv4/ruq8UpUtIlIqOSpe4jKXSyxky9AbPvgt3HpFMAAB0HcQ7PxA87WISGllzvOCB3O9PDgVFs3inX0t8suUoadMrHqSG+t+srszB2hrDipbrlyuzlxESuuoyUHf8/m5wf1HW7YDDlvXMqy1LfJLlKGnXFMzj35kTBujyhYRKbdFs4L7j4aM37YtuI1dBmXoKcPt3eiNVdkiIuUUNZdL//5BBJO5XHXoqWU+JHpjVbaISDlFzeVSUx25qTL0lJtbJ9NCxjdJlS0iUm4RFS/D2tojN1WGTjAgek3NPGoIDTRozhYRiYMOc7wESpGhJ6ZsMZyhT6x6khtrf0x/a969QfrMXJ25iMRBui96+DJob1GGHhbO0K+pmdexMwdVt4hI/CyaFUwSiDL0DsIZuqpbRCQRQn2S6tBDwhm6qltEJBFCfdL47dtVh54WztBvbp3MLs+I/1XdIiJxE6p2KUWGnphB0cw69Hbs41921n9fOOsmDYiKSLyk+6QF05Whh6Uz9HSFSz9rwSz1C681euIuEZFYaNulDD0snaGrwkVEEiV1NzVl6CHpDF0VLiKSKKm+SXXoIekMXRUuIpIoqb5JGXpIOkO/uXUyLa75W0QkIVKVLsrQQ9IZemP7ybzi9TR7Ne1ubK79pOZvEZH4Ss3rMr6tWhl6WjpDn1j1JCNtLbW08bZ/gtk+RZ25iMTbUZN5fOBgZehpb2/Z8XHJYq21Ywb1Ve9yTcu/wbJ55W6eiEh2y+bxTvOWyFW9NkOPKlnsb80qWRSReFs0Sxl62Kkjh6pkUUSSaeu66Dp0996boatkUUQSaVB9dB26We/N0KMm5drudSpZFJF4mzAzPnXoZnamma0ys9VmNiNifR8zuy+1/mkzG1G0Fqak69CbU/cQdYf3fS9urr1MVS4iEm9HTWZY3eDIVSXN0M2sGpgDnAWMBqaY2eiMzS4CNrv7YcCtwE1Fa2HKtP1e4MbaHzPQdqXaBX1pZvTwvYu9KxGRohu/92HRGXrf4UXbRz5n6McBq919jbs3A/cCkzK2mQT8Z+rxA8AEs4iCy24Y/9adkRUu49+6s5i7ERHpEY9vbIrO0Dc2FW0f+XToBwBrQ8/XpZZFbuPurcBW4BOZL2RmU82sycyaNm3alLm6S/t59Pb7eZbKFxGRGHknS2+bbfmeKOmgqLvPdfcGd28YOnRo7i8I2WjR22+0LJUvIiIxMqy9sOV7Ip8OfT1wYOh5fWpZ5DZmVgMMAt4rRgPT1o6dzg6v67Bsh9exduz0Yu5GRKRHTDv0PPq2d8zQ+7Y70w49r2j7yKdDfxY43MwOMbM64HygMWObRuDC1OO/Ax51j5iFphuOnXgxy8d9h3cYSrsb7zCU5eO+w7ETLy7mbkREesQ5p9zAdYecx/5tjrmzf5tz3SHncc4pNxRtH5ZPv2tmZwM/BKqBn7r7d81sFtDk7o1m1hf4OXAM8D5wvruv6eo1GxoavKmpeIMBIiK9gZk95+4NUevyukm0uy8AFmQsmxl6vBP4QncaKSIi3ZOYK0VFRKRr6tBFRCqEOnQRkQqhDl1EpELkVeXSIzs22wS8uYdfPgTobZeI6ph7Bx1z79CdYz7Y3SOvtCxbh94dZtaUrWynUumYewcdc+/QU8esyEVEpEKoQxcRqRBJ7dDnlrsBZaBj7h10zL1DjxxzIjN0ERHpLKln6CIikkEduohIhYh1hx6Hm1OXWh7HfJWZrTSzZWa2yMwOLkc7iynXMYe2+1szczNLfIlbPsdsZpNT7/UKM/tFqdtYbHn8bB9kZovN7IXUz/fZ5WhnsZjZT81so5ktz7LezOy21PdjmZmN7fZO3T2W/wim6v0TcChQB/wRGJ2xzWXAj1KPzwfuK3e7S3DMpwL9U48v7Q3HnNpuIPA4sBRoKHe7S/A+Hw68AOyTer5fudtdgmOeC1yaejwaeKPc7e7mMY8HxgLLs6w/G/g1YMAJwNPd3Wecz9BjcXPqEst5zO6+2N23p54uJbiDVJLl8z4D3ADcBOwsZeN6SD7H/BVgjrtvBnD3jSVuY7Hlc8wO7J16PAh4u4TtKzp3f5zg/hDZTALu9sBSYLCZ7d+dfca5Qy/azakTJJ9jDruI4Dd8kuU85tSfoge6+yOlbFgPyud9PgI4wsyeMrOlZnZmyVrXM/I55uuAC8xsHcH9F75WmqaVTaGf95zyusGFxI+ZXQA0AJ8qd1t6kplVAT8AvlzmppRaDUHscgrBX2GPm9lfufuWcjaqh00Bfubu3zezvwF+bmZj3L2It1GubHE+Q4/FzalLLJ9jxsxOB74JTHT3XSVqW0/JdcwDgTHAEjN7gyBrbEz4wGg+7/M6oNHdW9z9deBVgg4+qfI55ouAeQDu/gegL8EkVpUqr897IeLcocfi5tQllvOYzewY4C6CzjzpuSrkOGZ33+ruQ9x9hLuPIBg3mOjuSb4hbT4/2w8TnJ1jZkMIIpgu79Mbc/kc81vABAAzG0XQoW8qaStLqxH4n6lqlxOAre6+oVuvWO6R4ByjxGcTnJn8Cfhmatksgg80BG/4/cBq4Bng0HK3uQTH/Dvgz8CLqX+N5W5zTx9zxrZLSHiVS57vsxFETSuBlwhuvF72dvfwMY8GniKogHkR+HS529zN470H2AC0EPzFdRFwCXBJ6D2ek/p+vFSMn2td+i8iUiHiHLmIiEgB1KGLiFQIdegiIhVCHbqISIVQhy4iUiHUoYuIVAh16CIiFeL/AwzJAuJOQCgkAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for curve in [left, bottom, right, top]:\n",
    "    curve.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "You can (and need to) attach `BoundaryConditions` to each `BoundaryCurve` of a domain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> type(value)\n",
      "<class '__main__.EulerState'>\n",
      "ipdb> isinstance(value, State)\n",
      "True\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(248)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    247 \u001b[0;31m            \u001b[0;31m# field with the right value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 248 \u001b[0;31m            \u001b[0mbcs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    249 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(250)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    249 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 250 \u001b[0;31m            \u001b[0;32mfor\u001b[0m \u001b[0mfield\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfields\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    251 \u001b[0;31m                \u001b[0mbcs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfield\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(251)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    250 \u001b[0;31m            \u001b[0;32mfor\u001b[0m \u001b[0mfield\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfields\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 251 \u001b[0;31m                \u001b[0mbcs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfield\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    252 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(255)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    254 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 255 \u001b[0;31m        \u001b[0;32melif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    256 \u001b[0;31m            \u001b[0;31m# We transform the value to a callable that returns constant value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(257)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    256 \u001b[0;31m            \u001b[0;31m# We transform the value to a callable that returns constant value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 257 \u001b[0;31m            \u001b[0mset_value\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_SetValueCallable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    258 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(259)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    258 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 259 \u001b[0;31m        \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    260 \u001b[0;31m        \u001b[0mobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_value\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset_value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(260)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    259 \u001b[0;31m        \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 260 \u001b[0;31m        \u001b[0mobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_value\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset_value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    261 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(262)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    261 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 262 \u001b[0;31m        \u001b[0;32mreturn\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    263 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "--Return--\n",
      "<josie.bc.Dir...x7ff8f2539d30>\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(262)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    261 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 262 \u001b[0;31m        \u001b[0;32mreturn\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    263 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(250)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    249 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 250 \u001b[0;31m            \u001b[0;32mfor\u001b[0m \u001b[0mfield\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfields\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    251 \u001b[0;31m                \u001b[0mbcs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfield\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(251)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    250 \u001b[0;31m            \u001b[0;32mfor\u001b[0m \u001b[0mfield\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfields\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 251 \u001b[0;31m                \u001b[0mbcs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfield\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    252 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(255)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    254 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 255 \u001b[0;31m        \u001b[0;32melif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    256 \u001b[0;31m            \u001b[0;31m# We transform the value to a callable that returns constant value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(257)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    256 \u001b[0;31m            \u001b[0;31m# We transform the value to a callable that returns constant value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 257 \u001b[0;31m            \u001b[0mset_value\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_SetValueCallable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    258 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(259)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    258 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 259 \u001b[0;31m        \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    260 \u001b[0;31m        \u001b[0mobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_value\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset_value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(260)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    259 \u001b[0;31m        \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 260 \u001b[0;31m        \u001b[0mobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_value\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset_value\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    261 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ipdb> n\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(262)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    261 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 262 \u001b[0;31m        \u001b[0;32mreturn\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    263 \u001b[0;31m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n",
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> c\n"
     ]
    }
   ],
   "source": [
    "from josie.bc import Dirichlet, Neumann\n",
    "\n",
    "Q_zero = np.zeros(len(EulerState.fields)).view(EulerState)\n",
    "dQ_zero = Q_zero\n",
    "\n",
    "left.bc = Dirichlet(Q_zero)\n",
    "top.bc = Neumann(dQ_zero)\n",
    "bottom.bc = Dirichlet(Q_zero)\n",
    "right.bc = Dirichlet(Q_zero)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "You can also provide mixed Boundary conditions per each field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "> \u001b[0;32m/home/rdb/git/x/josiepy/josie/bc.py\u001b[0m(244)\u001b[0;36m__new__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m    243 \u001b[0;31m        \u001b[0m__import__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ipdb\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_trace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m--> 244 \u001b[0;31m        \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mState\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\u001b[0;32m    245 \u001b[0;31m            \u001b[0;31m# An entire state was provided as value of the Dirichlet BC, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0m\n",
      "ipdb> p value\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "from josie.bc import BoundaryCondition\n",
    "\n",
    "Q_bc = EulerState(rho=Dirichlet(1), rhoU=Neumann(0), rhoV=Dirichlet(1), \n",
    "                  rhoE=Dirichlet(1), rhoe=Dirichlet(3), U=Neumann(0), \n",
    "                  V=Neumann(0), p=Dirichlet(1), c=Dirichlet(10), e=Dirichlet(1))\n",
    "\n",
    "left.bc = BoundaryCondition(Q_bc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Then you can effectively generate the mesh object"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.mesh import Mesh\n",
    "from josie.mesh.cell import SimpleCell\n",
    "\n",
    "mesh = Mesh(left, bottom, right, top, SimpleCell)\n",
    "points = mesh.interpolate(20, 20)  # Gen. internal points\n",
    "mesh.generate() # Generate connectivity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "mesh.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PS: You can create 1D meshes, creating a rectangular mesh with just 1 cell in the vertical direction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The `Problem` object.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "\n",
    "\\begin{equation}\n",
    "    \\partial_t \\mathbf{q} + \n",
    "    \\nabla \\cdot \\left(\\underline{\\underline{\\mathbf{F}}}(\\mathbf{q}) \n",
    "        + \\underline{\\underline{\\underline{\\underline{D(\\mathbf{q})}}}} \\cdot \\nabla \\mathbf{q} \\right) +\n",
    "    \\underline{\\underline{\\underline{\\mathbf{B}(\\mathbf{q})}}} \\cdot \\nabla\\mathbf{q} = \n",
    "    \\mathbf{s}(\\mathbf{q})\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "`Problem` is the \"continuous\" representation of our problem. It implements the $\\mathbf{F}, \\mathbf{D}, \\mathbf{B}, \\mathbf{s}$ operators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.problem import Problem\n",
    "\n",
    "Problem??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.euler.problem import EulerProblem\n",
    "\n",
    "EulerProblem??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The `Scheme` object"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "It represents the discrete representation of our problem, after being discretized with Finite-Volume-Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "\\begin{split}\n",
    "    V_i\\partial_t \\mathbf{q}_i &+\n",
    "    \\sum_{f \\in \\text{faces}} \\left| \\mathbf{F} \\cdot \\hat{\\mathbf{n}} \\right|_f S_f + \n",
    "    \\sum_{f \\in \\text{faces}} \\left| \\left(\\mathbf{D}(\\mathbf{q})\\cdot\\nabla\\mathbf{q} \\right) \\cdot \\hat{\\mathbf{n}} \\right|_f S_f + \\\\\n",
    "    &+ \\left<\\mathbf{B}(\\mathbf{q})\\right>_i \\cdot \\sum_{f \\in \\text{faces}} \\left| \\mathbf{q}\\hat{\\mathbf{n}} \\right|_f S_f\n",
    "    - \\left< \\mathbf{s}(\\mathbf{q})\\right>_i V_i = \\mathbf{0}\n",
    "\\end{split}"
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    "Schemes are organized in spatial and time schemes. "
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    "### Spatial Schemes\n",
    "Spatial schemes implement the different discretizations for each operators _w.r.t._ **a single cell and its neighbour, whatever that neighbour is (no knowledge of left/right, top/bottom, etc...)**\n",
    "\n",
    "![cell_scheme](data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8" standalone="no"?>
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)\n",
    "\n",
    "and are organized in specific classes: "
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   "source": [
    "from josie.scheme.nonconservative import NonConservativeScheme\n",
    "from josie.scheme.convective import ConvectiveScheme\n",
    "from josie.scheme.source import SourceScheme\n",
    "# from josie.scheme.diffusion import DiffusionScheme <to be implemented>\n"
   ]
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    "#### `ConvectiveScheme`\n",
    "\n",
    "![cell_scheme](data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8" standalone="no"?>
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    "\n"
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    "It implements, in its `ConvectiveScheme.F` method, the discretization:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathbf{F}_\\frac{1}{2}(\\mathbf{q})= \\left| \\mathbf{F} \\cdot \\hat{\\mathbf{n}} \\right|_f S_f\n",
    "\\end{equation}\n",
    "\n",
    "on **one face**"
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   "source": [
    "ConvectiveScheme.F??"
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    "#### Example `Rusanov` for Euler system\n",
    "\n",
    "![cell_scheme](data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8" standalone="no"?>
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    "\n"
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    "It implements, in its `Rusanov.F` method, the discretization:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathbf{F}_\\frac{1}{2}(\\mathbf{q})= \n",
    "        \\frac{1}{2}\\left[\n",
    "            \\left(\\mathbf{F}(\\mathbf{q}_i) + \\mathbf{F}(\\mathbf{q}_{i+1})\\right) \\cdot \\hat{\\mathbf{n}} - \n",
    "            \\left|\\text{max}(\\lambda_p)\\right|\\left( \\mathbf{q}_{i+1} - \\mathbf{q}_i\\right)\n",
    "        \\right]\n",
    "\\end{equation}\n",
    "\n",
    "on **one face**"
   ]
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   "source": [
    "from josie.euler.schemes import Rusanov\n",
    "\n",
    "Rusanov.F??"
   ]
  },
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    "#### `NonConservativeScheme`\n",
    "\n",
    "![cell_scheme](data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8" standalone="no"?>
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    "\n"
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    "A discretization of the type: \n",
    "\n",
    "\\begin{equation}\n",
    "    \\left<\\mathbf{B}(\\mathbf{q})\\right>_i \\cdot \\sum_{f \\in \\text{faces}} \\left| \\mathbf{q}\\hat{\\mathbf{n}} \\right|_f S_f\n",
    "\\end{equation}\n",
    "\n",
    "is assumed, and the `NonConservativeScheme.G` implements the term:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathbf{G}_{\\frac{1}{2}}(\\mathbf{q}) = \\left| \\mathbf{q}\\hat{\\mathbf{n}} \\right|_f S_f\n",
    "\\end{equation}"
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    "#### Example `Upwind` for Baer-Nunziato non-conservative term\n",
    "\n",
    "![cell_scheme](data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8" standalone="no"?>
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)\n",
    "\n"
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   "source": [
    "It implements, in its `Upwind.G` method, the discretization:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathbf{\\alpha}_\\frac{1}{2}(\\mathbf{u}_i) \\hat{\\mathbf{n}} = \n",
    "    \\left[\n",
    "        \\left(\\mathbf{u}_{i_f} \\cdot \\hat{\\mathbf{n}} > 0\\right) \\alpha_i + \n",
    "        \\left(\\mathbf{u}_{i_f} \\cdot \\hat{\\mathbf{n}} < 0\\right) \\alpha_{i+1}\n",
    "    \\right] \\hat{\\mathbf{n}}\n",
    "\\end{equation}\n",
    "\n",
    "on **one face**"
   ]
  },
  {
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   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.bn.schemes import Upwind\n",
    "\n",
    "Upwind.G??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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    }
   },
   "source": [
    "### Time schemes\n",
    "Time schemes simply implement the time update for the discretized equation:\n",
    "\n",
    "\\begin{split}\n",
    "    V_i\\partial_t \\mathbf{q}_i &+\n",
    "    \\sum_{f \\in \\text{faces}} \\left| \\mathbf{F} \\cdot \\hat{\\mathbf{n}} \\right|_f S_f + \n",
    "    \\sum_{f \\in \\text{faces}} \\left| \\left(\\mathbf{D}(\\mathbf{q})\\cdot\\nabla\\mathbf{q} \\right) \\cdot \\hat{\\mathbf{n}} \\right|_f S_f + \\\\\n",
    "    &+ \\left<\\mathbf{B}(\\mathbf{q})\\right>_i \\cdot \\sum_{f \\in \\text{faces}} \\left| \\mathbf{q}\\hat{\\mathbf{n}} \\right|_f S_f\n",
    "    - \\left< \\mathbf{s}(\\mathbf{q})\\right>_i V_i = \\mathbf{0}\n",
    "\\end{split}\n",
    "\n",
    "In compact form:\n",
    "\\begin{equation}\n",
    "    \\partial_t \\mathbf{q}_i = - \\frac{1}{V_i}\\mathbf{f}\\left(\\mathbf{q}_i, t\\right)\n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathbf{f}$ contains all the discretized space operators. Integrating in time:\n",
    "\n",
    "\\begin{equation}\n",
    "    q_i^{k+1} = q_i^k - \\frac{1}{V_i}\\int_t^{t + \\Delta t} \\mathbf{f}\\left(\\mathbf{q}_i, \\tilde{t}\\right)\\; \\text{d}t\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   "source": [
    "`TimeScheme`\n",
    "\n",
    "It implements the term $\\frac{1}{V_i}\\int_t^{t+\\Delta t} \\mathbf{f}\\left(\\mathbf{q}_i, \\tilde{t}\\right)\\; \\text{d}t$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from josie.scheme.time import TimeScheme\n",
    "\n",
    "TimeScheme??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Example `ExplicitEuler`\n",
    "\n",
    "\\begin{equation}\n",
    "    \\frac{1}{V_i} \\int_t^{t + \\Delta t} \\mathbf{f}\\left(\\mathbf{q}_i, \\tilde{t}\\right)\\; \\text{d}t \\sim\n",
    "    \\frac{1}{V_i}\\mathbf{f}\\left(\\mathbf{q}_i^k, t_k\\right) \\Delta t\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from josie.general.schemes.time import ExplicitEuler\n",
    "\n",
    "ExplicitEuler??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": [
     "skip_conversion"
    ]
   },
   "source": [
    "## Let's implement a full solver for the Advection Equation\n",
    "\n",
    "<a href=\"002_Advection.ipynb\"> Click Here </a>"
   ]
  }
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