A high-order Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for scalar nonlinear conservation laws
<|reference_start|>A high-order Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for scalar nonlinear conservation laws: 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.<|reference_end|>
@article{chen2024a,
    title={A high-order Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV)
  method for scalar nonlinear conservation laws},
    author={Jiajie Chen, Joseph Nakao, Jing-Mei Qiu, Yang Yang},
    journal={arXiv preprint arXiv:2405.09835},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09835},
    primaryClass={math.NA cs.NA}
}