A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows

TL;DR

This paper introduces a high-order, cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows, significantly reducing numerical diffusion while maintaining simplicity and efficiency.

Contribution

It develops a 4th order accurate remapping correction for the cell-centered semi-Lagrangian scheme, improving accuracy without increasing complexity.

Findings

Achieves 4th order spatial accuracy in gas flow simulations.

Retains conservation and positivity properties of the original scheme.

Demonstrates effectiveness in 1D, 2D, and 3D Boltzmann equation problems.

Abstract

The term `Convected Scheme' (CS) refers to a family of algorithms, most usually applied to the solution of Boltzmann's equation, which uses a method of characteristics in an integral form to project an initial cell forward to a group of final cells. As such the CS is a `forward-trajectory' semi-Lagrangian scheme. For multi-dimensional simulations of neutral gas flows, the cell-centered version of this semi-Lagrangian (CCSL) scheme has advantages over other options due to its implementation simplicity, low memory requirements, and easier treatment of boundary conditions. The main drawback of the CCSL-CS to date has been its high numerical diffusion in physical space, because of the 2$^{\text{nd}}$ order remapping that takes place at the end of each time step. By means of a Modified Equation Analysis, it is shown that a high order estimate of the remapping error can be obtained a priori, and a small correction to the final position of the cells can be applied upon remapping, in order to achieve full compensation of this error. The resulting scheme is 4$^{\text{th}}$ order accurate in space while retaining the desirable properties of the CS: it is conservative and positivity-preserving, and the overall algorithm complexity is not appreciably increased. Two monotone (i.e. non-oscillating) versions of the fourth order CCSL-CS are also presented: one uses a common flux-limiter approach; the other uses a non-polynomial reconstruction to evaluate the derivatives of the density function. The method is illustrated in simple one- and two-dimensional examples, and a fully 3D solution of the Boltzmann equation describing expansion of a gas into vacuum through a cylindrical tube.