def test_linearized_mms_ldg_matrix_independence_from_dg_solution():
g = x_functions.Sine(offset=2.0)
r = -4.0 * np.power(np.pi, 2)
exact_solution = flux_functions.ExponentialFunction(g, r)
t_initial = 0.0
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 basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 4):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 20)
sine_dg = basis_.project(x_functions.Sine(offset=2.0), mesh_)
cosine_dg = basis_.project(x_functions.Cosine(offset=2.0),
mesh_)
tuple_ = problem.ldg_matrix(sine_dg, t_initial, bc, bc, bc, bc)
sine_matrix = tuple_[0]
sine_vector = tuple_[1]
tuple_ = problem.ldg_matrix(cosine_dg, t_initial, bc, bc, bc,
bc)
cosine_matrix = tuple_[0]
cosine_vector = tuple_[1]
assert np.linalg.norm(sine_matrix - cosine_matrix) <= tolerance
assert np.linalg.norm(sine_vector - cosine_vector) <= tolerance
def test_ldg_polynomials_convergence():
# LDG Diffusion should converge at 1st order for 1 basis_cpt
# or at num_basis_cpts - 4 for more basis_cpts
bc = boundary.Extrapolation()
t = 0.0
for i in range(4, 7):
hyper_diffusion.initial_condition = x_functions.Polynomial(degree=i)
exact_solution = hyper_diffusion.exact_time_derivative(
hyper_diffusion.initial_condition, t)
for num_basis_cpts in [1] + list(range(5, i + 1)):
for basis_class in basis.BASIS_LIST:
error_list = []
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(
hyper_diffusion.initial_condition, mesh_)
L = hyper_diffusion.ldg_operator(dg_solution, t, bc, bc,
bc, bc)
dg_error = math_utils.compute_dg_error(L, exact_solution)
error = dg_error.norm(slice(2, -2))
error_list.append(error)
# plot.plot_dg_1d(L, function=exact_solution, elem_slice=slice(1, -1))
order = utils.convergence_order(error_list)
# if already at machine precision don't check convergence
if error_list[-1] > tolerance:
if num_basis_cpts == 1:
assert order >= 1
else:
assert order >= num_basis_cpts - 4
def test_matrix_operator_equivalency():
t = 0.0
for bc in [boundary.Periodic(), boundary.Extrapolation()]:
for nonlinear_hyper_diffusion in test_problems:
nonlinear_hyper_diffusion.initial_condition = x_functions.Sine(
offset=2.0)
for num_basis_cpts in range(1, 6):
for basis_class in basis.BASIS_LIST:
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 20)
dg_solution = basis_.project(
nonlinear_hyper_diffusion.initial_condition, mesh_)
L = nonlinear_hyper_diffusion.ldg_operator(
dg_solution, t, bc, bc, bc, bc)
dg_vector = dg_solution.to_vector()
tuple_ = nonlinear_hyper_diffusion.ldg_matrix(
dg_solution, t, bc, bc, bc, bc)
matrix = tuple_[0]
vector = tuple_[1]
dg_error_vector = (L.to_vector() -
np.matmul(matrix, dg_vector) - vector)
dg_error = solution.DGSolution(dg_error_vector, basis_,
mesh_)
error = dg_error.norm() / L.norm()
# plot.plot_dg_1d(dg_error)
assert error <= tolerance
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_ldg_cos():
# LDG Diffusion should converge at 1st order for 1 basis_cpt
# or at num_basis_cpts - 4 for more basis_cpts
t = 0.0
bc = boundary.Periodic()
for nonlinear_hyper_diffusion in [hyper_diffusion_identity]:
nonlinear_hyper_diffusion.initial_condition = x_functions.Cosine(
offset=2.0)
exact_solution = nonlinear_hyper_diffusion.exact_time_derivative(
nonlinear_hyper_diffusion.initial_condition, t)
for num_basis_cpts in [1] + list(range(5, 6)):
for basis_class in basis.BASIS_LIST:
error_list = []
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(
nonlinear_hyper_diffusion.initial_condition, mesh_)
L = nonlinear_hyper_diffusion.ldg_operator(
dg_solution, t, bc, bc, bc, bc)
dg_error = math_utils.compute_dg_error(L, exact_solution)
error = dg_error.norm()
error_list.append(error)
# plot.plot_dg_1d(L, function=exact_solution)
order = utils.convergence_order(error_list)
# if already at machine precision don't check convergence
if error_list[-1] > tolerance:
if num_basis_cpts == 1:
assert order >= 1
else:
assert order >= num_basis_cpts - 4
def test_linearized_mms_ldg_convergence():
# LDG Diffusion should converge at 1st order for 1 basis_cpt
# or at num_basis_cpts - 2 for more basis_cpts
t = 0.0
bc = boundary.Periodic()
p_class = convection_hyper_diffusion.NonlinearHyperDiffusion
p_func = p_class.linearized_manufactured_solution
exact_solution = flux_functions.AdvectingSine(offset=2.0)
for diffusion_function in diffusion_functions:
problem = p_func(exact_solution, diffusion_function)
exact_time_derivative = problem.exact_time_derivative(
exact_solution, t)
for num_basis_cpts in [1] + list(range(5, 6)):
for basis_class in basis.BASIS_LIST:
error_list = []
basis_ = basis_class(num_basis_cpts)
# 10 and 20 elems maybe not in asymptotic regime yet
for num_elems in [20, 40]:
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems)
dg_solution = basis_.project(exact_solution, mesh_, t)
L = problem.ldg_operator(dg_solution, t, bc, bc)
dg_error = math_utils.compute_dg_error(
L, exact_time_derivative)
error = dg_error.norm()
error_list.append(error)
# plot.plot_dg_1d(L, function=exact_time_derivative)
order = utils.convergence_order(error_list)
# if already at machine precision don't check convergence
if error_list[-1] > tolerance:
if num_basis_cpts == 1:
assert order >= 1
else:
assert order >= num_basis_cpts - 4
def test_imex_linear_diffusion():
# advection with linear diffusion
# (q_t + q_x = q_xx + s(x, t))
exact_solution = xt_functions.AdvectingSine(offset=2.0)
problem = convection_diffusion.ConvectionDiffusion.manufactured_solution(
exact_solution)
t_initial = 0.0
bc = boundary.Periodic()
error_dict = dict()
cfl_list = [0.9, 0.3, 0.1]
for num_basis_cpts in range(1, 4):
imex = imex_runge_kutta.get_time_stepper(num_basis_cpts)
cfl = cfl_list[num_basis_cpts - 1]
# take 10 timesteps at coarsest time interval
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.BASIS_LIST:
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, include_source=False)
# this is a constant matrix case
(matrix, vector) = problem.ldg_matrix(dg_solution,
t_initial,
bc,
bc,
include_source=False)
solve_operator = time_stepping.get_solve_function_constant_matrix(
matrix, vector)
final_solution = time_stepping.time_step_loop_imex(
dg_solution,
t_initial,
t_final,
delta_t,
imex,
explicit_operator,
implicit_operator,
solve_operator,
)
dg_error = math_utils.compute_dg_error(final_solution,
exact_solution_final)
error = dg_error.norm()
error_list.append(error)
# plot.plot_dg_1d(final_solution, function=exact_solution_final)
# plot.plot_dg_1d(dg_error)
error_dict[num_basis_cpts] = error_list
order = utils.convergence_order(error_list)
assert order >= num_basis_cpts
def check_solution_operations_1d(basis_, tol2=tolerance):
mesh_ = mesh.Mesh1DUniform(-1.0, 1.0, 100)
# checking not inplace operations also checks inplace operations
# as not inplace operators refer to inplace operations
cos = x_functions.Cosine()
sin = x_functions.Sine()
cos_sol = basis_.project(cos, mesh_)
sin_sol = basis_.project(sin, mesh_)
# addition
def func(x):
return cos(x) + sin(x)
new_sol = basis_.do_solution_operation(cos_sol, sin_sol, operator.iadd)
projected_sol = basis_.project(func, mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
# subtraction
def func(x):
return cos(x) - sin(x)
new_sol = basis_.do_solution_operation(cos_sol, sin_sol, operator.isub)
projected_sol = basis_.project(func, mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
# multiplication, division, and power won't be exact for modal bases
# multiplication
def func(x):
return cos(x) * sin(x)
new_sol = basis_.do_solution_operation(cos_sol, sin_sol, operator.imul)
projected_sol = basis_.project(func, mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tol2
# division
def func(x):
return cos(x) / (sin(x) + 2)
sin_p2_sol = basis_.do_constant_operation(sin_sol, 2, operator.iadd)
new_sol = basis_.do_solution_operation(cos_sol, sin_p2_sol,
operator.itruediv)
projected_sol = basis_.project(func, mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tol2
# power
def func(x):
return (cos(x) + 2)**(sin(x) + 2)
cos_p2_sol = basis_.do_constant_operation(cos_sol, 2, operator.iadd)
new_sol = basis_.do_solution_operation(cos_p2_sol, sin_p2_sol,
operator.ipow)
projected_sol = basis_.project(func, mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tol2
def test_mesh1d_get_left_right_elems():
mesh1d_uniform = mesh.Mesh1DUniform(0.0, 1.0, 10)
for elem_index in range(mesh1d_uniform.num_elems):
assert mesh1d_uniform.get_left_elem_index(elem_index) == elem_index - 1
if elem_index < mesh1d_uniform.num_elems - 1:
assert mesh1d_uniform.get_right_elem_index(
elem_index) == elem_index + 1
else:
assert mesh1d_uniform.get_right_elem_index(elem_index) == -1
def test_compute_dg_error():
f = lambda x: np.cos(x)
for basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 3):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(f, mesh_)
dg_error = math_utils.compute_dg_error(dg_solution, f)
print(dg_error)
def test_diffusion_ldg_constant():
# LDG of one should be zero
diffusion.initial_condition = x_functions.Polynomial(degree=0)
t = 0.0
bc = boundary.Periodic()
for num_basis_cpts in range(1, 5):
for basis_class in basis.BASIS_LIST:
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(diffusion.initial_condition, mesh_)
L = diffusion.ldg_operator(dg_solution, t, bc, bc)
assert L.norm() <= tolerance
def test_legendre_limit_higher_moments_1d():
num_elems = 10
ic = x_functions.Sine()
limiting_constants = np.random.random_sample(num_elems)
legendre_basis = basis.LegendreBasis1D(3)
mesh_ = mesh.Mesh1DUniform(0, 1, num_elems)
dg_solution = legendre_basis.project(ic, mesh_)
initial_total_integral = dg_solution.total_integral()
limited_solution = legendre_basis.limit_higher_moments(
dg_solution, limiting_constants)
final_total_integral = limited_solution.total_integral()
assert initial_total_integral == final_total_integral
def test_derivative_dirichlet():
f = lambda q: 0.0
derivative_dirichlet = ldg_utils.DerivativeDirichlet(f, 1.0)
initial_condition = lambda x: np.cos(2.0 * np.pi * x)
initial_condition_integral = lambda x: np.zeros(x.shape)
for basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 4):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 20)
dg_solution = basis_.project(initial_condition, mesh_)
integral_dg_solution = basis_.project(initial_condition_integral,
mesh_)
dg_solution.integral = integral_dg_solution.coeffs
for i in mesh_.boundary_faces:
boundary_flux = derivative_dirichlet.evaluate_boundary(
dg_solution, i, None)
# should be same as interior value
# as integral satisfies boundary conditions
interior_value = dg_solution(mesh_.vertices[mesh_.faces[i]])
boundary_forcing = np.abs(boundary_flux - interior_value)
assert boundary_forcing <= tolerance
initial_condition = lambda x: np.cos(2.0 * np.pi * x)
initial_condition_integral = lambda x: np.ones(x.shape)
for basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 4):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 20)
dg_solution = basis_.project(initial_condition, mesh_)
integral_dg_solution = basis_.project(initial_condition_integral,
mesh_)
dg_solution.integral = integral_dg_solution.coeffs
for i in mesh_.boundary_faces:
boundary_flux = derivative_dirichlet.evaluate_boundary(
dg_solution, i, None)
# should be differenct from interior value
# as integral does not satisfies boundary conditions
interior_value = dg_solution(mesh_.vertices[mesh_.faces[i]])
boundary_forcing = np.abs(boundary_flux - interior_value)
assert boundary_forcing >= tolerance
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_get_faces_to_elems():
mesh1d_uniform = mesh.Mesh1DUniform(0.0, 1.0, 10)
mesh2d_cartesian = mesh.Mesh2DCartesian(0.0, 1.0, 0.0, 1.0, 10, 10)
mesh2d_unstructured = mesh.Mesh2DUnstructuredRectangle(
0.0, 1.0, 0.0, 1.0, 10, 10)
mesh_array = [mesh1d_uniform, mesh2d_cartesian, mesh2d_unstructured]
for mesh_ in mesh_array:
faces_to_elems = mesh_._get_faces_to_elems(mesh_.faces, mesh_.elems,
mesh_.boundary_faces)
for i_face in range(mesh_.num_faces):
for i_elem in faces_to_elems[i_face]:
assert math_utils.isin(i_elem, mesh_.face_to_elems[i_face])
def test_get_elems_to_faces():
mesh1d_uniform = mesh.Mesh1DUniform(0.0, 1.0, 10)
mesh2d_cartesian = mesh.Mesh2DCartesian(0.0, 1.0, 0.0, 1.0, 10, 10)
mesh2d_unstructured = mesh.Mesh2DUnstructuredRectangle(
0.0, 1.0, 0.0, 1.0, 10, 10)
mesh_array = [mesh1d_uniform, mesh2d_cartesian, mesh2d_unstructured]
for mesh_ in mesh_array:
elems_to_faces = mesh_._get_elems_to_faces(mesh_.faces, mesh_.elems,
mesh_.num_faces_per_elem)
for i_elem in range(mesh_.num_elems):
for i_face in elems_to_faces[i_elem]:
assert math_utils.isin(i_face, mesh_.elems_to_faces[i_elem])
def test_diffusion_ldg_polynomials_zero():
# LDG Diffusion of x should be zero in interior
bc = boundary.Extrapolation()
diffusion.initial_condition = x_functions.Polynomial(degree=1)
t = 0.0
for num_basis_cpts in range(1, 5):
for basis_class in basis.BASIS_LIST:
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(diffusion.initial_condition, mesh_)
L = diffusion.ldg_operator(dg_solution, t, bc, bc)
error = np.linalg.norm(L.coeffs[1:-1, :])
# plot.plot_dg_1d(L, elem_slice=slice(1, -1))
assert error < tolerance
def test_mesh1d_transform_to_mesh():
mesh1d_uniform = mesh.Mesh1DUniform(0.0, 1.0, 10)
for i in range(mesh1d_uniform.num_elems):
left_vertex_index = mesh1d_uniform.elems[i, 0]
left_vertex = mesh1d_uniform.vertices[left_vertex_index]
x = mesh1d_uniform.transform_to_mesh(-1.0, i)
assert np.abs(x - left_vertex) <= tolerance
right_vertex_index = mesh1d_uniform.elems[i, 1]
right_vertex = mesh1d_uniform.vertices[right_vertex_index]
x = mesh1d_uniform.transform_to_mesh(1.0, i)
assert np.abs(x - right_vertex) <= tolerance
elem_center = mesh1d_uniform.get_elem_center(i)
x = mesh1d_uniform.transform_to_mesh(0.0, i)
assert np.abs(elem_center - x) <= tolerance
def test_mesh1d_transform_to_canonical():
mesh1d_uniform = mesh.Mesh1DUniform(0.0, 1.0, 10)
for i in range(mesh1d_uniform.num_elems):
left_vertex_index = mesh1d_uniform.elems[i, 0]
left_vertex = mesh1d_uniform.vertices[left_vertex_index]
xi = mesh1d_uniform.transform_to_canonical(left_vertex, i)
assert np.abs(xi + 1.0) <= tolerance
right_vertex_index = mesh1d_uniform.elems[i, 1]
right_vertex = mesh1d_uniform.vertices[right_vertex_index]
xi = mesh1d_uniform.transform_to_canonical(right_vertex, i)
assert np.abs(xi - 1.0) <= tolerance
elem_center = mesh1d_uniform.get_elem_center(i)
xi = mesh1d_uniform.transform_to_canonical(elem_center)
assert np.abs(xi) <= tolerance
def check_constant_operations_1d(basis_):
mesh_ = mesh.Mesh1DUniform(-1.0, 1.0, 10)
coeffs = np.ones(basis_.space_order)
dg_solution = basis_.project(x_functions.Polynomial(coeffs), mesh_)
constant = 2.0
# addition
new_sol = basis_.do_constant_operation(dg_solution, constant,
operator.iadd)
new_coeffs = coeffs.copy()
new_coeffs[0] += constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
# subtraction
new_sol = basis_.do_constant_operation(dg_solution, constant,
operator.isub)
new_coeffs = coeffs.copy()
new_coeffs[0] -= constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
# multiplication
new_sol = basis_.do_constant_operation(dg_solution, constant,
operator.imul)
new_coeffs = coeffs * constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
# division
new_sol = basis_.do_constant_operation(dg_solution, constant,
operator.itruediv)
new_coeffs = coeffs / constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
# power - squaring
deg = basis_.num_basis_cpts // 2
dg_solution = basis_.project(x_functions.Polynomial(degree=deg), mesh_)
new_sol = basis_.do_constant_operation(dg_solution, 2.0, operator.ipow)
new_deg = deg * 2
projected_sol = basis_.project(x_functions.Polynomial(degree=new_deg),
mesh_)
error = np.linalg.norm(new_sol.coeffs - projected_sol.coeffs)
assert error <= tolerance
def test_compute_quadrature_matrix_one():
# quadrature_matrix should be same as M^{-1} S^T
t = 0.0
f = flux_functions.Polynomial(degree=0)
initial_condition = functions.Sine()
for basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 6):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(initial_condition, mesh_)
quadrature_matrix = ldg_utils.compute_quadrature_matrix(
dg_solution, t, f)
error = quadrature_matrix - basis_.mass_inverse_stiffness_transpose
assert np.linalg.norm(error) <= tolerance
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 test_mesh1d_uniform():
mesh1d_uniform = mesh.Mesh1DUniform(0.0, 1.0, 10)
assert mesh1d_uniform.num_elems == 10
assert mesh1d_uniform.num_vertices == 11
for i in range(mesh1d_uniform.num_elems):
assert mesh1d_uniform.elem_volumes[i] == 0.1
x = np.random.rand(10)
for xi in x:
elem_index = mesh1d_uniform.get_elem_index(xi)
elem = mesh1d_uniform.elems[elem_index]
vertex_1 = mesh1d_uniform.vertices[elem[0]]
vertex_2 = mesh1d_uniform.vertices[elem[1]]
assert xi >= np.min([vertex_1, vertex_2])
assert xi <= np.max([vertex_1, vertex_2])
def test_ldg_operator_constant():
# LDG of one should be zero
thin_film_diffusion.initial_condition = x_functions.Polynomial(degree=0)
t = 0.0
for bc in [boundary.Periodic(), boundary.Extrapolation()]:
for basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 5):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(
thin_film_diffusion.initial_condition, mesh_
)
L = thin_film_diffusion.ldg_operator(dg_solution, t, bc, bc, bc, bc)
# plot.plot_dg_1d(L)
assert L.norm() <= tolerance
def test_ldg_constant():
# LDG of one should be zero
t = 0.0
for nonlinear_hyper_diffusion in test_problems:
nonlinear_hyper_diffusion.initial_condition = x_functions.Polynomial(
degree=0)
for bc in [boundary.Periodic(), boundary.Extrapolation()]:
for num_basis_cpts in range(1, 5):
for basis_class in basis.BASIS_LIST:
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(
nonlinear_hyper_diffusion.initial_condition, mesh_)
L = nonlinear_hyper_diffusion.ldg_operator(
dg_solution, t, bc, bc, bc, bc)
assert L.norm() <= tolerance
def test_neumann():
boundary_derivative_function = flux_functions.Zero()
bc = boundary.Neumann(boundary_derivative_function)
basis_ = basis.LegendreBasis1D(3)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
f = x_functions.Sine()
dg_solution = basis_.project(f, mesh_)
problem = smooth_scalar_example.SmoothScalarExample(1.0, f)
riemann_solver = riemann_solvers.LocalLaxFriedrichs(problem)
boundary_faces = mesh_.boundary_faces
t = 0.0
flux = bc.evaluate_boundary(dg_solution, boundary_faces[0], riemann_solver,
t)
assert flux is not None
def test_ldg_constant():
# LDG discretization of 1 should be zero
hyper_diffusion.initial_condition = x_functions.Polynomial(degree=0)
t = 0.0
for bc in [boundary.Periodic(), boundary.Extrapolation()]:
for basis_class in basis.BASIS_LIST:
for num_basis_cpts in range(1, 6):
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(hyper_diffusion.initial_condition,
mesh_)
L = hyper_diffusion.ldg_operator(dg_solution, t, bc, bc, bc,
bc)
# plot.plot_dg_1d(L)
error = L.norm()
# quadrature_error can add up in higher basis_cpts
assert error <= 1e-4
def test_ldg_operator_equal_matrix():
f = x_functions.Sine()
t = 0.0
for bc in [boundary.Periodic(), boundary.Extrapolation()]:
for num_basis_cpts in range(1, 6):
for basis_class in basis.BASIS_LIST:
basis_ = basis_class(num_basis_cpts)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
dg_solution = basis_.project(f, mesh_)
L = diffusion.ldg_operator(dg_solution, t, bc, bc)
dg_vector = dg_solution.to_vector()
tuple_ = diffusion.ldg_matrix(dg_solution, t, bc, bc)
matrix = tuple_[0]
vector = tuple_[1]
error = np.linalg.norm(L.to_vector() -
np.matmul(matrix, dg_vector) - vector)
assert error <= tolerance
def test_interior():
bc = boundary.Extrapolation()
basis_ = basis.LegendreBasis1D(3)
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, 10)
f = x_functions.Sine()
dg_solution = basis_.project(f, mesh_)
problem = smooth_scalar_example.SmoothScalarExample(1.0, f)
riemann_solver = riemann_solvers.LocalLaxFriedrichs(problem)
boundary_faces = mesh_.boundary_faces
t = 0.0
flux = bc.evaluate_boundary(dg_solution, boundary_faces[0], riemann_solver,
t)
x = mesh_.get_face_position(boundary_faces[0])
assert flux == dg_solution(x)