A high-order Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for scalar nonlinear conservation laws

TL;DR

The paper develops a forward Eulerian-Lagrangian Runge-Kutta finite volume method (EL-RK-FV) that solves scalar hyperbolic conservation laws with shocks by tracing forward approximate characteristics and merging intersecting regions, enabling time steps well beyond standard CFL limits. It combines high-order SSP Runge-Kutta time integration with nonuniform ENO/WENO-AO spatial reconstructions and a merging procedure to handle shock-formed intersections, with Strang splitting employed for extending to 2D. The approach sharpens post-shock solutions while preserving stability (TVD/MPP in the base case) and demonstrates high-order convergence and robust shock/case capture in Burgers' equation through extensive 1D and 2D tests. Overall, the method offers a scalable, high-order framework for nonlinear scalar conservation laws that can extend to multi-dimensional problems and more general flux functions, while relaxing time-step restrictions inherent in classical Eulerian schemes.

Abstract

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.

Figures (17)

• Figure 2.1: The backward EL-RK-FV space-time region.
• Figure 3.2: The forward EL-RK-FV space-time region.
• Figure 3.3: A visualization of the five types of troubled cells defined in Definition \ref{['defn: troubledcell']}.
• Figure 3.4: (a) Two troubled cells over the stencil $\{I_{j-1},I_j,I_{j+1}\}$ for which the effective troubled cell that determines the influence region is $I_j$. (b) Step 3, the merging of two overlapping influence regions from two neighboring effective troubled cells are not isolated.
• Figure 3.5: The merging of the cells in the influence region of an isolated effective troubled cell; see case 4 in Definition \ref{['defn: influenceregion']}.

Theorems & Definitions (19)

• Remark 2.1
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Remark 3.4
• Definition 3.5
• Remark 3.6
• Remark 3.7
• Example 4.1
• Example 4.2