Semi-Lagrangian finite difference method for fluid and kinetic applications

Abstract

High order schemes are essential components in scientific computing area due to their superior properties, such as high efficiency and high resolution. In this PhD dissertation, we aim to investigate efficient, high order accurate and stable numerical methods. In particular, we propose a new mass conservative semi-Lagrangian (SL) finite difference (FD) method for solving linear advection equation, and apply the new scheme to the nonlinear Vlasov-Poisson (VP) system and the multi-scale BGK model. ☐ The first part of this work includes developing an efficient high order SLFD scheme for linear transport equations, which are further applied to the Vlasov simulations. Compared with the Eulerian type of schemes, the SL schemes allow extra large time stepping sizes without stability issue, which greatly improve the computational efficiency. For the nonlinear VP system, we apply the proposed SLFD scheme under the Strang splitting framework. The performance of the method is showcased by a group of numerical tests in one and two dimensions. ☐ In the second part, we use the proposed SLFD method, coupled with the diagonally implicit Runge Kutta (DIRK) time integration methods on the BGK model in both kinetic and the limiting fluid regime. In addition, we analyze the numerical stability of the fully discretized SLFD-DIRK scheme along characteristics. The Fourier stability analysis is performed through the Von Neumann analysis to a stiff two-velocity hyperbolic relaxation system. Asymptotic accuracy and asymptotic preserving properties of DIRK methods as well as various DIRK Butcher tables are discussed in theoretical and computational aspects. Extensive numerical tests are presented to verify the high order spatial and temporal accuracy, mass conservation, asymptotic accuracy and shock capturing capabilities of the proposed scheme.