Performance evaluations on the parallel CHAracteristic-Spectral-Mixed (CHASM) scheme

TL;DR

This paper thoroughly evaluates the parallel CHASM scheme for 6-D problems, demonstrating its accuracy, stability, and advantages over other methods in solving complex kinetic equations.

Contribution

It provides a detailed comparison of the CHASM scheme with other techniques, highlighting its effectiveness and improvements in handling high-dimensional kinetic equations.

Findings

Hermite boundary conditions improve spline approximation accuracy

Truncated kernel method effectively handles singular pseudodifferential operators

One-stage Lawson scheme outperforms multi-stage and splitting schemes in accuracy and stability

Abstract

Performance evaluations on the deterministic algorithms for 6-D problems are rarely found in literatures except some recent advances in the Vlasov and Boltzmann community [Dimarco et al. (2018), Kormann et al. (2019)], due to the extremely high complexity. Thus a detailed comparison among various techniques shall be useful to the researchers in the related fields. We try to make a thorough evaluation on a parallel CHAracteristic-Spectral-Mixed (CHASM) scheme to support its usage. CHASM utilizes the cubic B-spline expansion in the spatial space and spectral expansion in the momentum space, which many potentially overcome the computational burden in solving classical and quantum kinetic equations in 6-D phase space. Our purpose is three-pronged. First, we would like show that by imposing some effective Hermite boundary conditions, the local cubic spline can approximate to the global one as accurately as possible. Second, we will illustrate the necessity of adopting the truncated kernel method in calculating the pseudodifferential operator with a singular symbol, since the widely used pseudo-spectral method [Ringhofer (1990)] might fail to properly tackle the singularity. Finally, we make a comparison among non-splitting Lawson schemes and Strang operator splitting. Our numerical results demonstrate the advantage of the one-stage Lawson predictor-corrector scheme over multi-stage ones as well as the splitting scheme in both accuracy and stability.