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.

翻译：摘要：本文提出一类高阶欧拉-拉格朗日龙格-库塔有限体积方法，可数值求解含激波形成的Burgers方程，并推广至一般标量守恒律。欧拉-拉格朗日（EL）与半拉格朗日（SL）方法近年来发展迅速，已成为允许大时间步长的标准方法。然而，在激波形成后保持相对较大的时间步长仍颇具挑战。本文所提方案将偏微分方程在由Rankine-Hugoniot跳跃条件确定的特征线线性近似划分的时空区域上积分。我们沿时间正向追踪特征线，并提出网格单元合并流程以处理激波导致的特征线相交。基于此划分，将方程写成时间微分形式，并以线法方式结合龙格-库塔方法推进。空间重构采用ENO和WENO-AO等高分辨率方法。高维扩展通过维度分裂实现。数值实验表明，该方法具有高阶精度，并能在采用大时间步长时清晰地捕捉激波后解。