A high-order Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for scalar nonlinear conservation laws

TL;DR

This paper introduces a high-order Eulerian-Lagrangian Runge-Kutta finite volume method for scalar conservation laws, effectively capturing shocks with large time steps and high accuracy.

Contribution

The paper develops a novel high-order EL-RK-FV scheme that handles shocks and large time steps by integrating PDEs over space-time regions and merging intersecting mesh cells.

Findings

Achieves high-order accuracy in numerical experiments.

Effectively captures shocks with large time steps.

Demonstrates extension to higher dimensions.

Abstract

We present a class of high-order Eulerian-Lagrangian Runge-Kutta finite volume methods that can numerically solve Burgers' equation with shock formations, which could be extended to general scalar conservation laws. Eulerian-Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine-Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge-Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme's high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes.