Fourth-order conservative non-splitting semi-Lagrangian Hermite WENO schemes for kinetic and fluid simulations

TL;DR

This paper introduces fourth-order conservative semi-Lagrangian Hermite WENO schemes that achieve high accuracy and mass conservation for kinetic and fluid equations, enabling larger time steps and improved computational efficiency.

Contribution

The paper develops novel fourth-order semi-Lagrangian HWENO schemes with non-splitting, mass conservation, and positivity-preserving features for complex kinetic and fluid systems.

Findings

Achieved fourth-order accuracy in space and time.

Demonstrated large time step capability with high accuracy.

Validated effectiveness through extensive benchmark tests.

Abstract

We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass conservation naturally holds due to the weak formulation of the characteristic Galerkin method and the design of the HWENO reconstructions. We apply a positive-preserving limiter to maintain the positivity of numerical solutions when needed. Although the proposed SL framework allows us to take large time steps for improving computational efficiency, it also brings challenges to the spatial reconstruction technique; we construct two kind of novel HWENO reconstructions to fit the need for the proposed SL framework. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.