Non-splitting Eulerian-Lagrangian WENO schemes for two-dimensional nonlinear convection-diffusion equations

TL;DR

This paper introduces high-order, conservative Eulerian-Lagrangian WENO schemes for 2D nonlinear convection-diffusion equations, improving accuracy and stability by evolving solutions along modified characteristic lines with efficient reconstruction techniques.

Contribution

The paper presents a novel non-splitting Eulerian-Lagrangian Runge-Kutta finite volume WENO scheme that handles complex reconstructions on moving meshes for convection-diffusion equations.

Findings

Ensures mass conservation through flux-form semi-discretization.

Achieves high-order accuracy with efficient WENO reconstruction.

Successfully verifies effectiveness via extensive numerical tests.

Abstract

In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme.