discrete1.hybrid1d module

Hybrid multi-group transport solver for 1D time-dependent problems.

This module provides hybrid multi-group methods that automatically coarsen energy groups and/or angular discretization for improved computational efficiency in time-dependent 1D transport problems.

discrete1.hybrid1d.auto_bdf1(flux_last, xs_total_u, xs_scatter_u, xs_fission_u, velocity_u, external_u, boundary_u, medium_map, delta_x, angle_xu, angle_wu, bc_x, angles_c, groups_c, energy_grid, steps, dt, chi_u=None, chi_c=None, geometry=1)

Hybrid solver with automatic coarsening using first-order time integration.

Automatically coarsens energy groups and/or angular discretization for improved efficiency, then solves using backward Euler (BDF1) time stepping.

Parameters:

• flux_last (numpy.ndarray) – Angular flux at previous time (cells, angles, groups).

• xs_total_u (numpy.ndarray) – Fine-group total cross section (cells, groups_u).

• xs_scatter_u (numpy.ndarray) – Fine-group scattering cross section (cells, groups_u).

• xs_fission_u (numpy.ndarray) – Fine-group fission cross section (cells, groups_u).

• velocity_u (numpy.ndarray) – Fine-group velocity (groups_u,).

• external_u (numpy.ndarray) – Fine-group external source (steps/1, cells, angles, groups_u).

• boundary_u (numpy.ndarray) – Fine-group boundary conditions (steps/1, 2, angles, groups_u).

• medium_map (numpy.ndarray) – Material region map (cells,).

• delta_x (numpy.ndarray) – Cell widths (cells,).

• angle_xu (numpy.ndarray) – Fine angular discretization (angles_u,).

• angle_wu (numpy.ndarray) – Fine quadrature weights (angles_u,).

• bc_x (int) – Boundary condition type (0=vacuum, 1=reflective).

• angles_c (int) – Target coarse angular discretization.

• groups_c (int) – Target coarse energy groups.

• energy_grid (tuple) – Energy grid specification (edges, indices_u, indices_c).

• steps (int) – Number of time steps.

• dt (float) – Time step size.

• geometry (int, optional) – Geometry type (1=slab, 2=cylindrical, 3=spherical).

Returns:

Scalar flux at all time steps (steps, cells, groups_u).

Return type:

numpy.ndarray

discrete1.hybrid1d.auto_bdf2(flux_last_1, xs_total_u, xs_scatter_u, xs_fission_u, velocity_u, external_u, boundary_u, medium_map, delta_x, angle_xu, angle_wu, bc_x, angles_c, groups_c, energy_grid, steps, dt, chi_u=None, chi_c=None, geometry=1)

Hybrid solver with automatic coarsening using second-order time integration.

Automatically coarsens energy groups and/or angular discretization for improved efficiency, then solves using second-order backward differentiation formula (BDF2) time stepping.

Parameters:

• flux_last_1 (numpy.ndarray) – Angular flux at previous time (cells, angles, groups).

• xs_total_u (numpy.ndarray) – Fine-group total cross section (cells, groups_u).

• xs_scatter_u (numpy.ndarray) – Fine-group scattering cross section (cells, groups_u).

• xs_fission_u (numpy.ndarray) – Fine-group fission cross section (cells, groups_u).

• velocity_u (numpy.ndarray) – Fine-group velocity (groups_u,).

• external_u (numpy.ndarray) – Fine-group external source (steps/1, cells, angles, groups_u).

• boundary_u (numpy.ndarray) – Fine-group boundary conditions (steps/1, 2, angles, groups_u).

• medium_map (numpy.ndarray) – Material region map (cells,).

• delta_x (numpy.ndarray) – Cell widths (cells,).

• angle_xu (numpy.ndarray) – Fine angular discretization (angles_u,).

• angle_wu (numpy.ndarray) – Fine quadrature weights (angles_u,).

• bc_x (int) – Boundary condition type (0=vacuum, 1=reflective).

• angles_c (int) – Target coarse angular discretization.

• groups_c (int) – Target coarse energy groups.

• energy_grid (tuple) – Energy grid specification (edges, indices_u, indices_c).

• steps (int) – Number of time steps.

• dt (float) – Time step size.

• geometry (int, optional) – Geometry type (1=slab, 2=cylindrical, 3=spherical).

Returns:

Scalar flux at all time steps (steps, cells, groups_u).

Return type:

numpy.ndarray

discrete1.hybrid1d.backward_euler(flux_last, xs_total_u, xs_total_c, xs_scatter_u, xs_scatter_c, xs_fission_u, xs_fission_c, velocity_u, velocity_c, external_u, boundary_u, medium_map, delta_x, angle_xu, angle_xc, angle_wu, angle_wc, bc_x, fine_idx, coarse_idx, factor, steps, dt, chi_u=None, chi_c=None, geometry=1)

Hybrid solver with explicit coarsening using first-order time integration.

Solves using two independent energy group systems (uncollided and collided) and recombines with first-order backward Euler time stepping.

Parameters:

• flux_last (numpy.ndarray) – Angular flux at previous time (cells, angles_u, groups_u).

• xs_total_u (numpy.ndarray) – Fine-group total cross section (cells, groups_u).

• xs_total_c (numpy.ndarray) – Coarse-group total cross section (cells, groups_c).

• xs_scatter_u (numpy.ndarray) – Fine-group scattering (cells, groups_u).

• xs_scatter_c (numpy.ndarray) – Coarse-group scattering (cells, groups_c).

• xs_fission_u (numpy.ndarray) – Fine-group fission (cells, groups_u).

• xs_fission_c (numpy.ndarray) – Coarse-group fission (cells, groups_c).

• velocity_u (numpy.ndarray) – Fine-group velocity (groups_u,).

• velocity_c (numpy.ndarray) – Coarse-group velocity (groups_c,).

• external_u (numpy.ndarray) – Fine-group external source (steps/1, cells, angles_u, groups_u).

• boundary_u (numpy.ndarray) – Fine-group boundary (steps/1, 2, angles_u, groups_u).

• medium_map (numpy.ndarray) – Material region map (cells,).

• delta_x (numpy.ndarray) – Cell widths (cells,).

• angle_xu (numpy.ndarray) – Fine angular discretization (angles_u,).

• angle_xc (numpy.ndarray) – Coarse angular discretization (angles_c,).

• angle_wu (numpy.ndarray) – Fine quadrature weights (angles_u,).

• angle_wc (numpy.ndarray) – Coarse quadrature weights (angles_c,).

• bc_x (int) – Boundary condition type (0=vacuum, 1=reflective).

• fine_idx (numpy.ndarray) – Mapping from coarse to fine groups.

• coarse_idx (numpy.ndarray) – Mapping from fine to coarse groups.

• factor (float) – Energy group coarsening factor.

• steps (int) – Number of time steps.

• dt (float) – Time step size.

• geometry (int, optional) – Geometry type (1=slab, 2=cylindrical, 3=spherical).

Returns:

Scalar flux at all time steps (steps, cells, groups_u).

Return type:

numpy.ndarray

discrete1.hybrid1d.bdf2(flux_last_1, xs_total_u, xs_total_c, xs_scatter_u, xs_scatter_c, xs_fission_u, xs_fission_c, velocity_u, velocity_c, external_u, boundary_u, medium_map, delta_x, angle_xu, angle_xc, angle_wu, angle_wc, bc_x, fine_idx, coarse_idx, factor, steps, dt, chi_u=None, chi_c=None, geometry=1)

Hybrid solver with explicit coarsening using second-order time integration.

Solves using two independent energy group systems (uncollided and collided) and recombines with second-order backward differentiation formula (BDF2) time stepping.

Parameters:

• flux_last_1 (numpy.ndarray) – Angular flux at previous time (cells, angles_u, groups_u).

• xs_total_u (numpy.ndarray) – Fine-group total cross section (cells, groups_u).

• xs_total_c (numpy.ndarray) – Coarse-group total cross section (cells, groups_c).

• xs_scatter_u (numpy.ndarray) – Fine-group scattering (cells, groups_u).

• xs_scatter_c (numpy.ndarray) – Coarse-group scattering (cells, groups_c).

• xs_fission_u (numpy.ndarray) – Fine-group fission (cells, groups_u).

• xs_fission_c (numpy.ndarray) – Coarse-group fission (cells, groups_c).

• velocity_u (numpy.ndarray) – Fine-group velocity (groups_u,).

• velocity_c (numpy.ndarray) – Coarse-group velocity (groups_c,).

• external_u (numpy.ndarray) – Fine-group external source (steps/1, cells, angles_u, groups_u).

• boundary_u (numpy.ndarray) – Fine-group boundary (steps/1, 2, angles_u, groups_u).

• medium_map (numpy.ndarray) – Material region map (cells,).

• delta_x (numpy.ndarray) – Cell widths (cells,).

• angle_xu (numpy.ndarray) – Fine angular discretization (angles_u,).

• angle_xc (numpy.ndarray) – Coarse angular discretization (angles_c,).

• angle_wu (numpy.ndarray) – Fine quadrature weights (angles_u,).

• angle_wc (numpy.ndarray) – Coarse quadrature weights (angles_c,).

• bc_x (int) – Boundary condition type (0=vacuum, 1=reflective).

• fine_idx (numpy.ndarray) – Mapping from coarse to fine groups.

• coarse_idx (numpy.ndarray) – Mapping from fine to coarse groups.

• factor (float) – Energy group coarsening factor.

• steps (int) – Number of time steps.

• dt (float) – Time step size.

• geometry (int, optional) – Geometry type (1=slab, 2=cylindrical, 3=spherical).

Returns:

Scalar flux at all time steps (steps, cells, groups_u).

Return type:

numpy.ndarray