def test_nu_BDF4_diff(resolution, spatial_order):
grid = field.UniformPeriodicGrid(resolution, 50)
x = grid.values
IC = np.exp(-(x - 30)**2 / 4)
u = field.Field(grid, IC)
target = 1 / np.sqrt(5) * np.exp(-(x - 30)**2 / 20)
d = spatial.FiniteDifferenceUniformGrid(2,
spatial_order,
grid,
stencil_type='centered')
ts = timesteppers.BackwardDifferentiationFormula(u, d, 4)
alpha = 0.2
ts.evolve(1, alpha * grid.dx / 2)
ts.evolve(3, alpha * grid.dx / 3)
ts.evolve(4, alpha * grid.dx)
error = np.max(np.abs(u.data - target))
error_est = error_nu_BDF4_diff[(resolution, spatial_order)]
assert error < error_est
def test_nu_BDF5_wave(resolution, spatial_order):
grid = field.UniformPeriodicGrid(resolution, 2 * np.pi)
x = grid.values
IC = np.exp(-(x - np.pi)**2 * 8)
u = field.Field(grid, IC)
target = np.exp(-(x - np.pi + 2 * np.pi * 0.2)**2 * 8)
d = spatial.FiniteDifferenceUniformGrid(1,
spatial_order,
grid,
stencil_type='centered')
ts = timesteppers.BackwardDifferentiationFormula(u, d, 5)
alpha = 0.5
num_periods = 1.2
ts.evolve(0.5 * np.pi * num_periods, alpha * grid.dx)
ts.evolve(1.25 * np.pi * num_periods, alpha * grid.dx / 3)
ts.evolve(2 * np.pi * num_periods, alpha * grid.dx / 2)
error = np.max(np.abs(u.data - target))
error_est = error_nu_BDF5_wave[(resolution, spatial_order)]
assert error < error_est
def test_BDF6_wave(resolution):
grid = field.UniformPeriodicGrid(resolution, 2 * np.pi)
x = grid.values
IC = np.exp(-(x - np.pi)**2 * 8)
u = field.Field(grid, IC)
target = np.exp(-(x - np.pi - 2 * np.pi * 0.2)**2 * 8)
d = spatial.FiniteDifferenceUniformGrid(1,
6,
grid,
stencil_type='centered')
ts = timesteppers.BackwardDifferentiationFormula(u, d, 6)
alpha = 0.5
num_periods = 1.8
ts.evolve(2 * np.pi * num_periods, alpha * grid.dx)
error = np.max(np.abs(u.data - target))
error_est = error_BDF6_wave[resolution]
assert error < error_est