A high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting

TL;DR

This paper extends a high-order semi-Lagrangian discontinuous Galerkin method to efficiently solve 2D incompressible Euler equations and guiding center Vlasov models, avoiding operator splitting and incorporating adaptive time-stepping.

Contribution

It introduces a novel high-order semi-Lagrangian DG method for complex 2D models without operator splitting, with an adaptive time-stepping strategy to handle upstream cell distortion.

Findings

Successfully applied to 2D Euler and Vlasov models

Achieved stable large time steps without cell distortion

Enhanced accuracy with high-order characteristics tracing

Abstract

In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017) to the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation and the guiding center Vlasov model. We adopt a local DG method for Poisson's equation of these models. For tracing the characteristics, we adopt a high order characteristics tracing mechanism based on a prediction-correction technique. The SLDG with large time-stepping size might be subject to extreme distortion of upstream cells. To avoid this problem, we propose a novel adaptive time-stepping strategy by controlling the relative deviation of areas of upstream cells.