BGK1D1V

BGK1D1V is the one-dimensional-in-space, one-dimensional-in-velocity BGK model. In the implementation, the local relaxation time is reconstructed as

\[\tau(x,t) = \frac{\tilde{\tau}}{n(x,t)\sqrt{T(x,t)}}\]

rather than being prescribed directly as a field.

State and quadrature

State1D1V stores:

  • f, f_tmp, f_m
  • uniform grids x, v
  • macroscopic moments n, u, T
  • cached masks and coefficients used by transport kernels

Moment evaluation is performed by direct discrete quadrature,

\[n = \int f\,dv, \qquad \nu = \int vf\,dv, \qquad u = \frac{\nu}{n}, \qquad T = \frac{1}{n}\int v^2 f\,dv - u^2.\]

Boundary handling

All current BGK1D steppers keep the boundary distribution fixed at each time step.

  • explicit copies the left and right boundary cells after the update;
  • implicit caches f_bc_l and f_bc_r and keeps them fixed during Picard iteration;
  • holo overwrites the boundary cells after the HO solve.

This is therefore not a periodic implementation.

Backend support

Scheme torch cuda_kernel cpu_kernel
explicit supported supported not registered
implicit not registered supported supported
holo not registered supported not registered

Explicit stepper

The explicit BGK1D stepper performs:

  1. moment evaluation,
  2. Maxwellian reconstruction,
  3. local relaxation-time construction,
  4. first-order upwind transport,
  5. collision update,
  6. boundary fixation and buffer swap.

The torch version is a transparent reference implementation; the cuda_kernel version delegates the heavy work to a fused extension.

Implicit stepper

The implicit stepper is implemented in both CUDA and CPU extension variants. The algorithmic structure is the same in both cases.

  1. copy the current distribution into the implicit workspace,
  2. compute initial moments W=(n, \nu, T),
  3. optionally inject a prescribed initial W or a CNN-based warm-start,
  4. build the tridiagonal coefficients from the current moments,
  5. solve the batched tridiagonal systems along the spatial direction,
  6. recompute moments and test convergence,
  7. update W directly or through Anderson acceleration.

The convergence metric can be chosen as either distribution-based (conv_type="f") or moment-based (conv_type="w").

CNN warm-start

When moments_cnn_modelpath is set, the implicit stepper loads a checkpoint and predicts moment increments before Picard iteration begins. The current implementation supports:

  • primitive or conservative input channels,
  • optional temporal-history augmentation through prev_delta,
  • dw and dnu output conventions,
  • gradient-based attenuation through warm_delta_weight_mode="w_grad".

HOLO stepper

The HOLO implementation uses an outer high-order iteration and an inner low-order moment solve.

  • The HO layer computes face moments and a heat-flux-like term Q from the current distribution.
  • The LO layer solves a 3x3 block-tridiagonal system for W=(n, \nu, U).
  • The distribution update is then obtained from a tridiagonal solve using the LO moments.

The implementation uses theta=0.5, i.e. a Crank-Nicolson-like split.

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