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fluxes.py
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fluxes.py
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import cupy as cp
import variables as var
def basis_product(flux, basis_arr, axis, permutation):
return cp.transpose(cp.tensordot(flux, basis_arr,
axes=([axis], [1])),
axes=permutation)
class DGFlux:
def __init__(self, resolutions, order, grid, nu):
self.x_ele, self.y_ele, self.u_res, self.v_res = resolutions
self.x_res = grid.x.wavenumbers.shape[0]
self.y_res = grid.y.wavenumbers.shape[0]
self.order = order
# permutations after tensor-dot with basis array
self.permutations = [(0, 1, 2, 5, 3, 4), # for contraction with u nodes
(0, 1, 2, 3, 4, 5)] # for contraction with v nodes
# slices into the DG boundaries (list of tuples)
self.boundary_slices = [[(slice(self.x_res), slice(self.y_res),
slice(self.u_res), 0,
slice(self.v_res), slice(self.order)),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res), -1,
slice(self.v_res), slice(self.order))],
[(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(self.v_res), 0),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(self.v_res), -1)]]
self.boundary_slices_pad = [[(slice(self.x_res), slice(self.y_res),
slice(self.u_res + 2), 0,
slice(self.v_res), slice(self.order)),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res + 2), -1,
slice(self.v_res), slice(self.order))],
[(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(self.v_res + 2), 0),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(self.v_res + 2), -1)]]
self.flux_input_slices = [(slice(self.x_res), slice(self.y_res),
slice(1, self.u_res + 1), slice(self.order),
slice(self.v_res), slice(self.order)),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(1, self.v_res + 1), slice(self.order)),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(self.v_res), slice(self.order))]
self.pad_slices = [(slice(self.x_res), slice(self.y_res),
slice(1, self.u_res + 1),
slice(self.v_res), slice(self.order)),
(slice(self.x_res), slice(self.y_res),
slice(self.u_res), slice(self.order),
slice(1, self.v_res + 1))]
self.num_flux_sizes = [(self.x_res, self.y_res, self.u_res, 2, self.v_res, self.order),
(self.x_res, self.y_res, self.u_res, self.order, self.v_res, 2)]
self.padded_flux_sizes = [(self.x_res, self.y_res, self.u_res + 2, self.order, self.v_res, self.order),
(self.x_res, self.y_res, self.u_res, self.order, self.v_res + 2, self.order)]
self.directions = [2, 4]
self.sub_elements = [3, 5]
# arrays
self.flux_ex = var.Distribution(resolutions=resolutions, order=order)
self.flux_ey = var.Distribution(resolutions=resolutions, order=order)
self.flux_bz = var.Distribution(resolutions=resolutions, order=order)
# particle charge
self.charge = -1.0
self.nu = nu # hyper-viscosity
self.pad_field, self.pad_spectrum = None, None
def semi_discrete_fully_explicit(self, distribution, field, grid):
""" Computes the semi-discrete equation with a full advection step """
# Compute the flux (pseudospectral method)
self.flux_ex.arr_spectral = self.compute_flux(distribution=distribution, field=field.electric_x, grid=grid)
self.flux_ey.arr_spectral = self.compute_flux(distribution=distribution, field=field.electric_y, grid=grid)
self.flux_bz.arr_spectral = self.compute_flux(distribution=distribution, field=field.magnetic_z, grid=grid)
# Compute semi-discrete RHS
return (grid.u.J[None, None, :, None, None, None] * self.u_flux(distribution=distribution, grid=grid) +
grid.v.J[None, None, None, None, :, None] * self.v_flux(distribution=distribution, grid=grid) +
self.source_term(distribution=distribution, grid=grid))
def semi_discrete_semi_implicit(self, distribution, field, grid):
""" Computes the semi-discrete equation with a half advection step"""
# Compute the flux (pseudospectral method)
self.flux_ex.arr_spectral = self.compute_flux(distribution=distribution, field=field.electric_x, grid=grid)
self.flux_ey.arr_spectral = self.compute_flux(distribution=distribution, field=field.electric_y, grid=grid)
self.flux_bz.arr_spectral = self.compute_flux(distribution=distribution, field=field.magnetic_z, grid=grid)
# Compute semi-discrete RHS
return (grid.u.J[None, None, :, None, None, None] * self.u_flux(distribution=distribution, grid=grid) +
grid.v.J[None, None, None, None, :, None] * self.v_flux(distribution=distribution, grid=grid) -
self.nu * (grid.x.device_wavenumbers_fourth[:, None, None, None, None, None] +
grid.y.device_wavenumbers_fourth[None, :, None, None, None, None]) *
distribution.arr_spectral)
def initialize_zero_pad(self, grid):
self.pad_field = cp.zeros((grid.x.modes + 2 * grid.x.pad_width, grid.y.modes + grid.y.pad_width)) + 0j
self.pad_spectrum = cp.zeros((grid.x.modes + 2 * grid.x.pad_width, grid.y.modes + grid.y.pad_width,
grid.u.elements, grid.u.order,
grid.v.elements, grid.v.order)) + 0j
def compute_flux(self, distribution, field, grid):
""" Compute the flux convolution(field, distribution) using pseudospectral method """
# Dealias with two-thirds rule
self.pad_field[grid.x.pad_width:-grid.x.pad_width, :-grid.y.pad_width] = field.arr_spectral
self.pad_spectrum[grid.x.pad_width:-grid.x.pad_width, :-grid.y.pad_width, :, :, :, :] = (
distribution.arr_spectral
)
# inverse transform zero-padded array
return forward_distribution_transform(cp.multiply(
inverse_field_transform(self.pad_field)[:, :, None, None, None, None],
inverse_distribution_transform(self.pad_spectrum))
)[grid.x.pad_width:-grid.x.pad_width, :-grid.y.pad_width, :, :, :, :]
def u_flux(self, distribution, grid):
# # flux = self.charge * (self.flux_ex.arr_spectral + cp.einsum('rps,nmijrs->nmijrp',
# # grid.v.translation_matrix,
# # self.flux_bz.arr_spectral))
# flux = self.charge * (self.flux_ex.arr_spectral + (grid.v.device_arr[None, None, None, None, :, :] *
# self.flux_bz.arr_spectral))
# return (basis_product(flux=flux, basis_arr=grid.u.local_basis.internal,
# axis=3, permutation=self.permutations[0]) -
# self.numerical_flux(distribution=distribution,
# flux=flux, grid=grid, dim=0))
# flux = self.charge * (self.flux_ex.arr_spectral + cp.einsum('rps,nmijrs->nmijrp',
# grid.v.translation_matrix,
# self.flux_bz.arr_spectral))
return (basis_product(flux=self.charge * (self.flux_ex.arr_spectral + cp.einsum('rps,nmijrs->nmijrp',
grid.v.translation_matrix,
self.flux_bz.arr_spectral)),
basis_arr=grid.u.local_basis.internal,
axis=3, permutation=self.permutations[0]) -
self.numerical_flux(distribution=distribution,
flux=self.charge * (self.flux_ex.arr_spectral +
(grid.v.device_arr[None, None, None, None, :, :] *
self.flux_bz.arr_spectral)),
grid=grid, dim=0))
def v_flux(self, distribution, grid):
# flux = self.charge * (self.flux_ey.arr_spectral - cp.einsum('ijk,nmikrs->nmijrs',
# grid.u.translation_matrix,
# self.flux_bz.arr_spectral))
# flux = self.charge * (self.flux_ey.arr_spectral - (grid.u.device_arr[None, None, :, :, None, None] *
# self.flux_bz.arr_spectral))
# return (basis_product(flux=flux, basis_arr=grid.v.local_basis.internal,
# axis=5, permutation=self.permutations[1]) -
# self.numerical_flux(distribution=distribution,
# flux=flux, grid=grid, dim=1))
return (basis_product(flux=self.charge * (self.flux_ey.arr_spectral - cp.einsum('ijk,nmikrs->nmijrs',
grid.u.translation_matrix,
self.flux_bz.arr_spectral)),
basis_arr=grid.v.local_basis.internal,
axis=5, permutation=self.permutations[1]) -
self.numerical_flux(distribution=distribution,
flux=self.charge * (self.flux_ey.arr_spectral -
(grid.u.device_arr[None, None, :, :, None, None] *
self.flux_bz.arr_spectral)),
grid=grid, dim=1))
def source_term(self, distribution, grid):
return -1j * (cp.multiply(grid.x.device_wavenumbers[:, None, None, None, None, None],
cp.einsum('axb,ijabcd->ijaxcd',
grid.u.translation_matrix, distribution.arr_spectral)) +
cp.multiply(grid.y.device_wavenumbers[None, :, None, None, None, None],
cp.einsum('cxd,ijabcd->ijabcx',
grid.v.translation_matrix, distribution.arr_spectral)))
def numerical_flux(self, distribution, flux, grid, dim):
# Allocate
num_flux = cp.zeros(self.num_flux_sizes[dim]) + 0j
# set padded flux
padded_flux = cp.zeros(self.padded_flux_sizes[dim]) + 0j
padded_flux[self.flux_input_slices[dim]] = flux
# Compute a central flux
num_flux[self.boundary_slices[dim][0]] = -1.0 * (cp.roll(padded_flux[self.boundary_slices_pad[dim][1]],
shift=+1,
axis=self.directions[dim])[self.pad_slices[dim]] +
flux[self.boundary_slices[dim][0]]) / 2.0
num_flux[self.boundary_slices[dim][1]] = (cp.roll(padded_flux[self.boundary_slices_pad[dim][0]],
shift=-1,
axis=self.directions[dim])[self.pad_slices[dim]] +
flux[self.boundary_slices[dim][1]]) / 2.0
# re-use padded_flux array for padded_distribution
padded_flux[self.flux_input_slices[dim]] = distribution.arr_spectral
constant = cp.amax(cp.absolute(flux), axis=self.sub_elements[dim])
# Lax-Friedrichs flux
num_flux[self.boundary_slices[dim][0]] += -1.0 * cp.multiply(constant,
(cp.roll(
padded_flux[self.boundary_slices_pad[dim][1]],
shift=+1,
axis=self.directions[dim])[
self.pad_slices[dim]] -
distribution.arr_spectral[
self.boundary_slices[dim][0]]) / 2.0)
num_flux[self.boundary_slices[dim][1]] += -1.0 * cp.multiply(constant,
(cp.roll(
padded_flux[self.boundary_slices_pad[dim][0]],
shift=-1,
axis=self.directions[dim])[
self.pad_slices[dim]] -
distribution.arr_spectral[
self.boundary_slices[dim][1]]) / 2.0)
return basis_product(flux=num_flux, basis_arr=grid.v.local_basis.numerical,
axis=self.sub_elements[dim], permutation=self.permutations[dim])
class SpaceFlux:
def __init__(self, c):
# self.resolutions = resolutions
self.c = c
def faraday(self, dynamic_field, grid):
return -1j * (grid.x.device_wavenumbers[:, None] * dynamic_field.electric_y.arr_spectral -
grid.y.device_wavenumbers[None, :] * dynamic_field.electric_x.arr_spectral)
def ampere_y(self, distribution, dynamic_field, grid):
return ((self.c ** 2.0) * (-1j * grid.x.device_wavenumbers[:, None] * dynamic_field.magnetic_z.arr_spectral) -
grid.charge_sign * distribution.moment1.arr_spectral[1, :, :])
def ampere_x(self, distribution, dynamic_field, grid):
return ((self.c ** 2.0) * (1j * grid.y.device_wavenumbers[None, :] * dynamic_field.magnetic_z.arr_spectral) -
grid.charge_sign * distribution.moment1.arr_spectral[0, :, :])
# def inverse_field_transform(field, dim):
# return cp.fft.irfft2(cp.fft.fftshift(field[dim, :, :], axes=0), norm='forward')
def inverse_field_transform(field):
return cp.fft.irfft2(cp.fft.fftshift(field, axes=0), norm='forward')
def inverse_distribution_transform(distribution):
return cp.fft.irfft2(cp.fft.fftshift(distribution, axes=0), axes=(0, 1), norm='forward')
def forward_distribution_transform(nodal_array):
return cp.fft.fftshift(cp.fft.rfft2(nodal_array, axes=(0, 1), norm='forward'), axes=0)