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：本文提出了一类高阶欧拉-拉格朗日龙格-库塔有限体积方法，能够数值求解具有激波形成的Burgers方程，并可推广至一般标量守恒律。近年来，欧拉-拉格朗日（EL）和半拉格朗日（SL）方法得到了持续发展，并因其允许采用较大时间步长而成为主流数值方法。然而，在激波形成后仍保持相对较大的时间步长依然颇具挑战性。我们所提出的方案在由Rankine-Hugoniot跳跃条件确定的特征线线性近似所划分的时空区域上对偏微分方程进行积分。我们沿时间向前追踪特征线，并针对因激波导致的特征线相交问题提出了网格单元的合并处理流程。基于此划分，我们将方程改写为时间微分形式，并以线法（method-of-lines）的方式采用龙格-库塔方法进行时间演化。空间重构采用ENO和WENO-AO等高分辨率格式。通过维数分裂方法将方案推广至高维情形。数值实验验证了本方案的高阶精度及其在采用大时间步长时仍能清晰捕捉激波后解的能力。