Error Estimates for Discontinuous Galerkin Approximations to the Vlasov-Unsteady Stokes System

TL;DR

This paper establishes uniqueness for the 3D Vlasov-unsteady Stokes problem, introduces a conservative semi-discrete scheme coupling DG methods, and provides optimal error estimates validated by numerical experiments.

Contribution

It presents a novel mass and momentum conservative DG-based semi-discrete scheme for the coupled Vlasov-Stokes system with proven optimal error estimates.

Findings

The scheme is mass and momentum conservative.

Optimal error estimates are derived for smooth initial data.

Numerical experiments confirm theoretical error bounds.

Abstract

In the first part of this paper, uniqueness of strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is based on the coupling of discontinuous Galerkin approximations for the Vlasov and the Stokes equations for the 2D problem. The proposed method is both mass and momentum conservative. Based on a special projection and also the Stokes projection, optimal error estimates in the case of smooth compactly supported initial data are derived. Moreover, the generalization of error estimates to 3D problem is also indicated. Finally, based on time splitting algorithm, some numerical experiments are conducted whose results confirm our theoretical findings.