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_constant_operations():
coeffs = np.ones(basis_.num_basis_cpts)
dg_solution = basis_.project(x_functions.Polynomial(coeffs), mesh_)
constant = 2.0
# addition
new_coeffs = coeffs.copy()
new_coeffs[0] += constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution + constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
new_sol = constant + dg_solution
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# subtraction
new_coeffs = coeffs.copy()
new_coeffs[0] -= constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution - constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
new_coeffs = -1.0 * coeffs
new_coeffs[0] += constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = constant - dg_solution
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# multiplication
new_coeffs = coeffs * constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution * constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
new_sol = constant * dg_solution
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# division
new_sol = dg_solution / constant
new_coeffs = coeffs / constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# right division not implemented
# power - squaring
deg = basis_.num_basis_cpts // 2
dg_solution = basis_.project(x_functions.Polynomial(degree=deg), mesh_)
new_sol = dg_solution ** 2
new_deg = deg * 2
projected_sol = basis_.project(x_functions.Polynomial(degree=new_deg), mesh_)
error = (new_sol - projected_sol).norm()
assert error <= tolerance
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_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_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 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_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_constant_operations_inplace():
coeffs = np.ones(basis_.num_basis_cpts)
dg_solution = basis_.project(x_functions.Polynomial(coeffs), mesh_)
constant = 2.0
# addition
new_coeffs = coeffs.copy()
new_coeffs[0] += constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution.copy()
new_sol += constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# subtraction
new_coeffs = coeffs.copy()
new_coeffs[0] -= constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution.copy()
new_sol -= constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# multiplication
new_coeffs = coeffs * constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution.copy()
new_sol *= constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# division
new_coeffs = coeffs / constant
projected_sol = basis_.project(x_functions.Polynomial(new_coeffs), mesh_)
new_sol = dg_solution.copy()
new_sol /= constant
error = (new_sol - projected_sol).norm()
assert error <= tolerance
# power - squaring
deg = basis_.num_basis_cpts // 2
dg_solution = basis_.project(x_functions.Polynomial(degree=deg), mesh_)
new_deg = deg * 2
projected_sol = basis_.project(x_functions.Polynomial(degree=new_deg), mesh_)
new_sol = dg_solution.copy()
new_sol **= 2
error = (new_sol - projected_sol).norm()
assert error <= tolerance
def test_ldg_polynomial_zero():
# LDG of x, x^2, x^3 should be zero in interior
bc = boundary.Extrapolation()
for i in range(1, 4):
hyper_diffusion.initial_condition = x_functions.Polynomial(degree=i)
t = 0.0
# for 1 < num_basis_cpts <= i not enough information to compute derivatives
# get rounding errors
for num_basis_cpts in [1] + list(range(i + 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(hyper_diffusion.initial_condition,
mesh_)
L = hyper_diffusion.ldg_operator(dg_solution, t, bc, bc, bc,
bc)
error = L.norm(slice(2, -2))
# plot.plot_dg_1d(L, elem_slice=slice(2, -2))
assert error <= tolerance
def test_ldg_operator_polynomial_zero():
# LDG of x, x^2 should be zero in interior
t = 0.0
for n in range(1, 3):
thin_film_diffusion.initial_condition = x_functions.Polynomial(degree=n)
for bc in [boundary.Periodic(), boundary.Extrapolation()]:
for basis_class in basis.BASIS_LIST:
# for 1 < num_basis_cpts <= i not enough information
# to compute derivatives get rounding errors
for num_basis_cpts in [1] + list(range(n + 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)
error = L.norm(slice(2, -2))
# plot.plot_dg_1d(L, elem_slice=slice(-2, 2))
assert error <= tolerance
def test_diffusion_ldg_polynomials_exact():
# LDG Diffusion should be exactly second derivative of polynomials in the interior
bc = boundary.Extrapolation()
t = 0.0
# x^i should be exact for i+1 or more basis_cpts
for i in range(2, 5):
diffusion.initial_condition = x_functions.Polynomial(degree=i)
exact_solution = diffusion.exact_time_derivative(
diffusion.initial_condition, t)
for num_basis_cpts in range(i + 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(diffusion.initial_condition,
mesh_)
L = diffusion.ldg_operator(dg_solution, t, bc, bc)
dg_error = math_utils.compute_dg_error(L, exact_solution)
error = dg_error.norm(slice(1, -1))
# plot.plot_dg_1d(dg_error, elem_slice=slice(1, -1))
assert error < tolerance
def test_diffusion_ldg_polynomials_convergence():
# LDG Diffusion should converge at 1st order for 1 basis_cpt
# or at num_basis_cpts - 2 for more basis_cpts
bc = boundary.Extrapolation()
t = 0.0
for nonlinear_diffusion in test_problems:
d = nonlinear_diffusion.diffusion_function.degree
# having problems at i >= d with convergence rate
# still small error just not converging properly
# exact solution is grows rapidly as x increases in this situation
# error must larger at x = 1 then at x = 0
# could also not be in asymptotic regime
for i in range(1, d):
nonlinear_diffusion.initial_condition = x_functions.Polynomial(
degree=i)
exact_solution = nonlinear_diffusion.exact_time_derivative(
nonlinear_diffusion.initial_condition, t)
for num_basis_cpts in [1] + list(range(3, i + 1)):
for basis_class in basis.BASIS_LIST:
error_list = []
basis_ = basis_class(num_basis_cpts)
for num_elems in [30, 60]:
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems)
dg_solution = basis_.project(
nonlinear_diffusion.initial_condition, mesh_)
L = nonlinear_diffusion.ldg_operator(
dg_solution, t, bc, bc)
dg_error = math_utils.compute_dg_error(
L, exact_solution)
error = dg_error.norm(slice(1, -1))
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 - 2
def test_ldg_polynomials_zero():
# LDG HyperDiffusion should be zero in the interior x, x^2
bc = boundary.Extrapolation()
t = 0.0
# x^i should be exact for i+1 or more basis_cpts
# needs lot more basis components if f(q, x, t) = q^2 or q^3
for nonlinear_hyper_diffusion in test_problems:
for i in range(1, 3):
nonlinear_hyper_diffusion.initial_condition = x_functions.Polynomial(
degree=i)
for num_basis_cpts in range(i + 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(
nonlinear_hyper_diffusion.initial_condition, mesh_)
L = nonlinear_hyper_diffusion.ldg_operator(
dg_solution, t, bc, bc, bc, bc)
error = L.norm(slice(2, -2))
# plot.plot_dg_1d(L, function=exact_solution, elem_slice=slice(1, -1))
assert error < tolerance
def test_ldg_polynomials_exact():
# LDG HyperDiffusion should be exact for polynomials in the interior
bc = boundary.Extrapolation()
t = 0.0
# x^i should be exact for i+1 or more basis_cpts
# needs lot more basis components if f(q, x, t) = q^2
for nonlinear_hyper_diffusion in [hyper_diffusion_identity]:
for i in range(3, 5):
nonlinear_hyper_diffusion.initial_condition = x_functions.Polynomial(
degree=i)
exact_solution = nonlinear_hyper_diffusion.exact_time_derivative(
nonlinear_hyper_diffusion.initial_condition, t)
for num_basis_cpts in range(i + 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(
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(slice(2, -2))
# plot.plot_dg_1d(L, function=exact_solution, elem_slice=slice(1, -1))
assert error < 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
# having problems at i >= 3 with convergence rate
# still small error just not converging properly
for i in range(3, 5):
thin_film_diffusion.initial_condition = x_functions.Polynomial(degree=i)
thin_film_diffusion.initial_condition.set_coeff((1.0 / i), i)
exact_solution = thin_film_diffusion.exact_time_derivative(
thin_film_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 [40, 80]:
mesh_ = mesh.Mesh1DUniform(0.0, 1.0, num_elems)
dg_solution = basis_.project(
thin_film_diffusion.initial_condition, mesh_
)
L = thin_film_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 and error_list[0] > tolerance:
if num_basis_cpts == 1:
assert order >= 1
else:
assert order >= num_basis_cpts - 4
def test_polynomial():
for i in range(4):
x_function = x_functions.Polynomial(degree=i)
check_x_function(x_function)