  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.
