def test_evaluate_weak_form(): periodic_bc = boundary.Periodic() t = 0.0 for problem in test_problems: exact_time_derivative = problem.exact_time_derivative initial_time_derivative = x_functions.FrozenT(exact_time_derivative, 0.0) riemann_solver = riemann_solvers.LocalLaxFriedrichs(problem) for basis_class in basis.BASIS_LIST: for num_basis_cpts in range(1, 4): error_list = [] basis_ = basis_class(num_basis_cpts) for num_elems in [20, 40]: mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) dg_solution = basis_.project(initial_condition, mesh_) result = dg_utils.evaluate_weak_form( dg_solution, t, problem.app_.flux_function, riemann_solver, periodic_bc, problem.app_.nonconservative_function, problem.app_.regularization_path, problem.app_.source_function) error = math_utils.compute_error(result, initial_time_derivative) error_list.append(error) order = utils.convergence_order(error_list) if num_basis_cpts == 1: assert order >= 1 else: assert order >= num_basis_cpts - 1
def test_compute_error(): f = lambda x: np.cos(x) for basis_class in basis.BASIS_LIST: for num_basis_cpts in range(1, 3): errorList = [] basis_ = basis_class(num_basis_cpts) for num_elems in [10, 20]: mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) dg_solution = basis_.project(f, mesh_) error = math_utils.compute_error(dg_solution, f) errorList.append(error) order = utils.convergence_order(errorList) assert order >= num_basis_cpts
def test_linearized_mms_ldg_irk(): # g = functions.Sine(offset=2.0) # r = -1.0 # exact_solution = flux_functions.ExponentialFunction(g, r) exact_solution = flux_functions.AdvectingSine(offset=2.0) t_initial = 0.0 t_final = 0.1 exact_solution_final = lambda x: exact_solution(x, t_final) bc = boundary.Periodic() p_class = convection_hyper_diffusion.NonlinearHyperDiffusion p_func = p_class.linearized_manufactured_solution for diffusion_function in diffusion_functions: problem = p_func(exact_solution, diffusion_function) for num_basis_cpts in range(1, 3): irk = implicit_runge_kutta.get_time_stepper(num_basis_cpts) for basis_class in basis.BASIS_LIST: basis_ = basis_class(num_basis_cpts) error_list = [] for i in [1, 2]: if i == 1: delta_t = 0.01 num_elems = 20 else: delta_t = 0.005 num_elems = 40 mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) dg_solution = basis_.project(problem.initial_condition, mesh_) # time_dependent_matrix time does matter matrix_function = lambda t: problem.ldg_matrix( dg_solution, t, bc, bc, bc, bc) rhs_function = problem.get_implicit_operator( bc, bc, bc, bc) solve_function = time_stepping.get_solve_function_matrix( matrix_function) new_solution = time_stepping.time_step_loop_implicit( dg_solution, t_initial, t_final, delta_t, irk, rhs_function, solve_function, ) error = math_utils.compute_error(new_solution, exact_solution_final) error_list.append(error) # plot.plot_dg_1d(new_solution, function=exact_solution_final) order = utils.convergence_order(error_list) assert order >= num_basis_cpts
def test_nonlinear_mms_ldg_irk(): exact_solution = flux_functions.AdvectingSine(amplitude=0.1, offset=0.15) t_initial = 0.0 t_final = 0.1 exact_solution_final = lambda x: exact_solution(x, t_final) bc = boundary.Periodic() p_func = thin_film.ThinFilmDiffusion.manufactured_solution problem = p_func(exact_solution) for num_basis_cpts in range(1, 3): irk = implicit_runge_kutta.get_time_stepper(num_basis_cpts) cfl = 0.5 for basis_class in basis.BASIS_LIST: basis_ = basis_class(num_basis_cpts) error_list = [] n = 40 for num_elems in [n, 2 * n]: mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) delta_t = cfl * mesh_.delta_x / exact_solution.wavespeed dg_solution = basis_.project(problem.initial_condition, mesh_) # time_dependent_matrix time does matter matrix_function = lambda t, q: problem.ldg_matrix(q, t, bc, bc, bc, bc) rhs_function = problem.get_implicit_operator(bc, bc, bc, bc) solve_function = time_stepping.get_solve_function_picard( matrix_function, num_basis_cpts, num_elems * num_basis_cpts ) new_solution = time_stepping.time_step_loop_implicit( dg_solution, t_initial, t_final, delta_t, irk, rhs_function, solve_function, ) error = math_utils.compute_error(new_solution, exact_solution_final) error_list.append(error) # plot.plot_dg_1d(new_solution, function=exact_solution_final) with open("thin_film_nonlinear_irk_test.yml", "a") as file: dict_ = dict() subdict = dict() subdict["cfl"] = cfl subdict["n"] = n subdict["error0"] = float(error_list[0]) subdict["error1"] = float(error_list[1]) subdict["order"] = float(np.log2(error_list[0] / error_list[1])) dict_[num_basis_cpts] = subdict yaml.dump(dict_, file, default_flow_style=False) order = utils.convergence_order(error_list) assert order >= num_basis_cpts
def single_run( problem, basis_, mesh_, bc, t_final, delta_t, num_picard_iterations=None ): if num_picard_iterations is None: num_picard_iterations = basis_.num_basis_cpts imex = imex_runge_kutta.get_time_stepper(basis_.num_basis_cpts) dg_solution = basis_.project(problem.initial_condition, mesh_) # weak dg form with flux_function and source term explicit_operator = problem.get_explicit_operator(bc) # ldg discretization of diffusion_function implicit_operator = problem.get_implicit_operator( bc, bc, bc, bc, include_source=False ) matrix_function = lambda t, q: problem.ldg_matrix( q, t, bc, bc, bc, bc, include_source=False ) solve_operator = time_stepping.get_solve_function_picard( matrix_function, num_picard_iterations, mesh_.num_elems * basis_.num_basis_cpts ) error_dict = dict() # def after_step_hook(dg_solution, time): # error = math_utils.compute_error( # dg_solution, lambda x: problem.exact_solution(x, time) # ) # error_dict[round(time, 8)] = error final_solution = time_stepping.time_step_loop_imex( dg_solution, 0.0, t_final, delta_t, imex, explicit_operator, implicit_operator, solve_operator, # after_step_hook ) exact_solution_final = x_functions.FrozenT(problem.exact_solution, t_final) error = math_utils.compute_error(final_solution, exact_solution_final) return (final_solution, error, error_dict)
def test_ldg_matrix_irk(): diffusion = convection_diffusion.Diffusion.periodic_exact_solution() t_initial = 0.0 t_final = 0.1 bc = boundary.Periodic() basis_ = basis.LegendreBasis1D(1) exact_solution = lambda x: diffusion.exact_solution(x, t_final) for num_basis_cpts in range(1, 3): irk = implicit_runge_kutta.get_time_stepper(num_basis_cpts) for basis_class in basis.BASIS_LIST: basis_ = basis_class(num_basis_cpts) error_list = [] # constant matrix for i in [1, 2]: if i == 1: delta_t = 0.01 num_elems = 20 else: delta_t = 0.005 num_elems = 40 mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) dg_solution = basis_.project(diffusion.initial_condition, mesh_) # constant matrix time doesn't matter tuple_ = diffusion.ldg_matrix(dg_solution, t_initial, bc, bc) matrix = tuple_[0] # vector = tuple_[1] rhs_function = diffusion.get_implicit_operator(bc, bc) solve_function = time_stepping.get_solve_function_constant_matrix( matrix) new_solution = time_stepping.time_step_loop_implicit( dg_solution, t_initial, t_final, delta_t, irk, rhs_function, solve_function, ) error = math_utils.compute_error(new_solution, exact_solution) error_list.append(error) order = utils.convergence_order(error_list) assert order >= num_basis_cpts
def test_advection_one_time_step(): def initial_condition(x): return np.sin(2.0 * np.pi * x) advection_ = advection.Advection(initial_condition=initial_condition) riemann_solver = riemann_solvers.LocalLaxFriedrichs( advection_.flux_function, advection_.wavespeed_function ) explicit_time_stepper = explicit_runge_kutta.ForwardEuler() boundary_condition = boundary.Periodic() cfl = 1.0 for basis_class in basis.BASIS_LIST: basis_ = basis_class(1) error_list = [] for num_elems in [20, 40]: mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) dg_solution = basis_.project(advection_.initial_condition, mesh_) delta_t = dg_utils.get_delta_t(cfl, advection_.wavespeed, mesh_.delta_x) time_initial = 0.0 time_final = delta_t rhs_function = lambda time, q: dg_utils.dg_weak_formulation( q, advection_.flux_function, riemann_solver, boundary_condition ) final_solution = time_stepping.time_step_loop_explicit( dg_solution, time_initial, time_final, delta_t, explicit_time_stepper, rhs_function, ) error = math_utils.compute_error( final_solution, lambda x: advection_.exact_solution(x, time_final) ) error_list.append(error) order = utils.convergence_order(error_list) assert order >= 1
def basis_convergence( test_function, initial_condition, exact_solution, order_check_function, x_left=0.0, x_right=1.0, basis_list=basis.BASIS_LIST, basis_cpt_list=range(1, 5), ): for basis_class in basis_list: for num_basis_cpts in basis_cpt_list: basis_ = basis_class(num_basis_cpts) error_list = [] for num_elems in [20, 40]: mesh_ = mesh.Mesh1DUniform(x_left, x_right, num_elems) dg_solution = basis_.project(initial_condition, mesh_) result = test_function(dg_solution) error = math_utils.compute_error(result, exact_solution) error_list.append(error) order = convergence_order(error_list) assert order_check_function(order, num_basis_cpts)
def test_advection_operator(): # test that dg_operator acting on projected initial condition converges to # exact time derivative # will lose one order of accuracy for i in range(2): if i == 0: sin = x_functions.Sine() cos = x_functions.Cosine() initial_condition = x_functions.ComposedVector([sin, cos]) else: initial_condition = x_functions.Sine() wavespeed = 1.0 exact_solution = advection.ExactSolution(initial_condition, wavespeed) exact_time_derivative = advection.ExactTimeDerivative(exact_solution, wavespeed) initial_time_derivative = x_functions.FrozenT(exact_time_derivative, 0.0) app_ = advection.Advection(wavespeed) riemann_solver = riemann_solvers.LocalLaxFriedrichs(app_.flux_function) boundary_condition = boundary.Periodic() for basis_class in basis.BASIS_LIST: for num_basis_cpts in range(1, 5): basis_ = basis_class(num_basis_cpts) error_list = [] for num_elems in [20, 40]: mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) dg_sol = basis_.project(initial_condition, mesh_) dg_operator = app_.get_explicit_operator( riemann_solver, boundary_condition ) F = dg_operator(0.0, dg_sol) error = math_utils.compute_error(F, initial_time_derivative) error_list.append(error) order = utils.convergence_order(error_list) assert order >= max([1.0, num_basis_cpts - 1])
def test_dg_operator(): # test that dg_operator converges to exact_time_derivative in smooth case initial_condition = x_functions.Sine(offset=2.0) max_wavespeed = 3.0 problem = smooth_example.SmoothExample(initial_condition, max_wavespeed) riemann_solver = riemann_solvers.LocalLaxFriedrichs(problem) boundary_condition = boundary.Periodic() x_left = 0.0 x_right = 1.0 exact_time_derivative = burgers.ExactTimeDerivative( problem.initial_condition) exact_time_derivative_initial = x_functions.FrozenT( exact_time_derivative, 0.0) for basis_class in basis.BASIS_LIST: for num_basis_cpts in range(1, 5): basis_ = basis_class(num_basis_cpts) error_list = [] for num_elems in [40, 80]: mesh_ = mesh.Mesh1DUniform(x_left, x_right, num_elems) dg_solution = basis_.project(problem.initial_condition, mesh_) explicit_operator = problem.app_.get_explicit_operator( riemann_solver, boundary_condition) dg_time_derivative = explicit_operator(0.0, dg_solution) error = math_utils.compute_error( dg_time_derivative, exact_time_derivative_initial) error_list.append(error) order = test_utils.convergence_order(error_list) if num_basis_cpts > 1: assert order >= num_basis_cpts - 1 else: assert order >= 1
def test_imex_linearized_mms(): # advection with linearized diffusion # (q_t + q_x = (f(x, t) q_xx + s(x, t)) exact_solution = flux_functions.AdvectingSine(amplitude=0.1, offset=0.15) p_class = convection_hyper_diffusion.ConvectionHyperDiffusion p_func = p_class.linearized_manufactured_solution t_initial = 0.0 bc = boundary.Periodic() for diffusion_function in [squared]: problem = p_func(exact_solution, None, diffusion_function) cfl_list = [0.9, 0.15, 0.1] for num_basis_cpts in range(2, 4): imex = imex_runge_kutta.get_time_stepper(num_basis_cpts) cfl = cfl_list[num_basis_cpts - 1] n = 20 t_final = cfl * (1.0 / n) / exact_solution.wavespeed exact_solution_final = lambda x: exact_solution(x, t_final) for basis_class in [basis.LegendreBasis1D]: basis_ = basis_class(num_basis_cpts) error_list = [] for num_elems in [n, 2 * n]: mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems) delta_t = cfl * mesh_.delta_x / exact_solution.wavespeed dg_solution = basis_.project(problem.initial_condition, mesh_) # weak dg form with flux_function and source term explicit_operator = problem.get_explicit_operator(bc) # ldg discretization of diffusion_function implicit_operator = problem.get_implicit_operator( bc, bc, bc, bc, include_source=False) # this is a constant matrix case matrix_function = lambda t: problem.ldg_matrix( dg_solution, t, bc, bc, bc, bc, include_source=False) solve_operator = time_stepping.get_solve_function_matrix( matrix_function) final_solution = time_stepping.time_step_loop_imex( dg_solution, t_initial, t_final, delta_t, imex, explicit_operator, implicit_operator, solve_operator, ) error = math_utils.compute_error(final_solution, exact_solution_final) error_list.append(error) # plot.plot_dg_1d(final_solution, function=exact_solution_final) order = utils.convergence_order(error_list) with open("hyper_diffusion_linearized_mms_test.yml", "a") as file: dict_ = dict() subdict = dict() subdict["cfl"] = cfl subdict["n"] = n subdict["error0"] = float(error_list[0]) subdict["error1"] = float(error_list[1]) subdict["order"] = float( np.log2(error_list[0] / error_list[1])) dict_[num_basis_cpts] = subdict yaml.dump(dict_, file, default_flow_style=False) assert order >= num_basis_cpts
def after_step_hook(dg_solution, time): error = math_utils.compute_error( dg_solution, lambda x: problem.exact_solution(x, time)) error_dict[round(time, 8)] = error
def test_imex_nonlinear_mms(): wavenumber = 1.0 / 20.0 x_left = 0.0 x_right = 40.0 exact_solution = flux_functions.AdvectingSine( amplitude=0.1, wavenumber=wavenumber, offset=0.15 ) p_func = thin_film.ThinFilm.manufactured_solution t_initial = 0.0 bc = boundary.Periodic() problem = p_func(exact_solution) cfl_list = [0.5, 0.1, 0.1] n = 40 for num_basis_cpts in range(3, 4): imex = imex_runge_kutta.get_time_stepper(num_basis_cpts) cfl = cfl_list[num_basis_cpts - 1] t_final = 10 * cfl * ((x_right - x_left) / n) / exact_solution.wavespeed exact_solution_final = lambda x: exact_solution(x, t_final) for basis_class in [basis.LegendreBasis1D]: basis_ = basis_class(num_basis_cpts) error_list = [] for num_elems in [n, 2 * n]: mesh_ = mesh.Mesh1DUniform(x_left, x_right, num_elems) delta_t = cfl * mesh_.delta_x / exact_solution.wavespeed dg_solution = basis_.project(problem.initial_condition, mesh_) # weak dg form with flux_function and source term explicit_operator = problem.get_explicit_operator(bc) # ldg discretization of diffusion_function implicit_operator = problem.get_implicit_operator( bc, bc, bc, bc, include_source=False ) matrix_function = lambda t, q: problem.ldg_matrix( q, t, bc, bc, bc, bc, include_source=False ) solve_operator = time_stepping.get_solve_function_picard( matrix_function, num_basis_cpts, num_elems * num_basis_cpts ) final_solution = time_stepping.time_step_loop_imex( dg_solution, t_initial, t_final, delta_t, imex, explicit_operator, implicit_operator, solve_operator, ) error = math_utils.compute_error(final_solution, exact_solution_final) error_list.append(error) # plot.plot_dg_1d(final_solution, function=exact_solution_final) with open("thin_film_nonlinear_mms_test.yml", "a") as file: dict_ = dict() subdict = dict() subdict["cfl"] = cfl subdict["n"] = n subdict["error0"] = float(error_list[0]) subdict["error1"] = float(error_list[1]) subdict["order"] = float(np.log2(error_list[0] / error_list[1])) dict_[num_basis_cpts] = subdict yaml.dump(dict_, file, default_flow_style=False) order = utils.convergence_order(error_list) assert order >= num_basis_cpts