2024/05/16 by Jiajie Chen, Chen, Jiajie, Joseph Nakao +5
Earth and Planetary Sciences · Engineering · #65M08 #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA) #Tropical and Extratropical Cyclones Research
paper · pdf · doi:10.48550/arxiv.2405.09835
openalex publication_date 2024/05/16 · openalex created_date 2024/05/20 · openalex updated_date 2026/07/28
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.