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