Regularity theory and numerical algorithm for the fractional Klein-Kramers equation

TL;DR

This paper introduces a new fully discrete numerical scheme for the fractional Klein-Kramers equation, combining backward Euler convolution quadrature and local discontinuous Galerkin methods, with proven error estimates and numerical validation.

Contribution

The paper develops a novel fully discrete scheme for the fractional Klein-Kramers equation with rigorous error analysis and independence from temporal regularity.

Findings

Error estimates are established for the scheme.

Numerical results confirm the theoretical convergence.

Scheme effectively handles hypocoercivity challenges.

Abstract

Fractional Klein-Kramers equation can well describe subdiffusion in phase space. In this paper, we develop the fully discrete scheme for fractional Klein-Kramers equation based on the backward Euler convolution quadrature and local discontinuous Galerkin methods. Thanks to the obtained sharp regularity estimates in temporal and spatial directions after overcoming the hypocoercivity of the operator, the complete error analyses of the fully discrete scheme are built. % , the main challenge of which comes from the hypocoercivity of the operator. It's worth mentioning that the convergence of the provided scheme is independent of the temporal regularity of the exact solution. Finally, numerical results are proposed to verify the correctness of the theoretical results.