Improve Rosenbrock finite difference by swapping dt with J

Fixes SciML/DifferentialEquations.jl#773

test/interface/difftype_tests.jl

@@ -0,0 +1,67 @@
+using OrdinaryDiffEq, Test
+
+function fcn(du, u, p, t)
+    du[1] = u[1]^2 - 1 / t^4
+end
+
+function f_jac(J, y, P, t)
+    #-- numerical Jac by FD
+    y_thres = P
+    del = sqrt(eps(1.0))
+    n = length(y)
+    f0 = similar(y)
+    f1 = similar(y)
+    fcn(f0, y, P, t)
+    for i = 1:n
+        del_1 = del * max(abs(y[i]), y_thres)
+        y1 = copy(y)
+        y1[i] = y1[i] + del_1
+        fcn(f1, y1, P, t)
+        J[:, i] = (f1 - f0) / del_1
+    end
+end
+
+tspan = (1.0, 1.0e3);
+M = zeros(1, 1);
+
+println("--- AD ---")
+
+f = ODEFunction(fcn, mass_matrix = M)
+problem = ODEProblem(f, [-1.0], tspan);
+
+sol = solve(problem, Rodas4P2(), maxiters = Int(1e7), reltol = 1.0e-12, abstol = 1.0e-12);
+@test sol.destats.naccept < 6100
+
+println("--- FD central ---")
+
+sol = solve(problem, Rodas4P2(autodiff = false, diff_type = Val{:central}), maxiters = Int(1e7), reltol = 1.0e-12, abstol = 1.0e-12);
+@test sol.destats.naccept < 6100
+
+println("--- FD forward ---")
+
+sol = solve(problem, Rodas4P2(autodiff = false, diff_type = Val{:forward}), maxiters = Int(1e7), reltol = 1.0e-12, abstol = 1.0e-12);
+@test sol.destats.naccept < 6100
+
+println("--- FD forward, y_thres = 1 ---")
+
+y_thres = 1.0;
+f = ODEFunction(fcn, mass_matrix = M, jac = f_jac)
+problem = ODEProblem(f, [-1.0], tspan, y_thres);
+sol = solve(problem, Rodas4P2(autodiff = false, diff_type = Val{:forward}), maxiters = Int(1e7), reltol = 1.0e-12, abstol = 1.0e-12);
+@test sol.destats.naccept < 6100
+
+println("--- FD forward, y_thres = 1.0e-5 ---")
+
+y_thres = 1.0e-5;
+f = ODEFunction(fcn, mass_matrix = M, jac = f_jac)
+problem = ODEProblem(f, [-1.0], tspan, y_thres);
+sol = solve(problem, Rodas4P2(autodiff = false, diff_type = Val{:forward}), maxiters = Int(1e7), reltol = 1.0e-12, abstol = 1.0e-12);
+@test sol.destats.naccept < 6100
+
+println("--- FD forward, y_thres = sqrt(eps) ---")
+
+y_thres = sqrt(eps(1.0));
+f = ODEFunction(fcn, mass_matrix = M, jac = f_jac)
+problem = ODEProblem(f, [-1.0], tspan, y_thres);
+sol = solve(problem, Rodas4P2(autodiff = false, diff_type = Val{:forward}), maxiters = Int(1e7), reltol = 1.0e-12, abstol = 1.0e-12);
+@test sol.destats.naccept < 6100

test/runtests.jl

@@ -43,6 +43,7 @@ end
         @time @safetestset "No Recompile Tests" begin include("interface/norecompile.jl") end
         @time @safetestset "Linear Nonlinear Solver Tests" begin include("interface/linear_nonlinear_tests.jl") end
         @time @safetestset "Linear Solver Tests" begin include("interface/linear_solver_test.jl") end
+        @time @safetestset "Diff Type Tests" begin include("interface/difftype_tests.jl") end
         @time @safetestset "Linear Solver Split ODE Tests" begin include("interface/linear_solver_split_ode_test.jl") end
         @time @safetestset "Sparse Diff Tests" begin include("interface/sparsediff_tests.jl") end
         @time @safetestset "Enum Tests" begin include("interface/enums.jl") end