A high-order Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for scalar nonlinear conservation laws (Q6645940)

The paper explores advancements in computational schemes for solving scalar nonlinear conservation laws, particularly focusing on shock-dominated scenarios.\N\NThe study addresses the computational challenges associated with hyperbolic conservation laws, a class of equations widely used in modelling phenomena in physics, meteorology, fluid dynamics, and other fields. Such problems are characterised by the development of discontinuities, including shocks, even from smooth initial conditions. The time-stepping constraints imposed by the Courant-Friedrichs-Lewy (CFL) condition further complicate numerical solutions, especially for nonlinear cases where traditional methods face difficulties in achieving accuracy and stability.\N\NTo address these challenges, the authors propose a high-order Eulerian-Lagrangian Runge-Kutta finite volume method. The novelty of their approach lies in combining Eulerian frameworks, which support high spatial resolution, with Lagrangian methods that enable solution evolution along characteristics, thus relaxing CFL restrictions. By partitioning the space-time domain using the Rankine-Hugoniot jump condition, the method forward-traces the characteristics and incorporates a novel cell-merging procedure to handle intersecting characteristics due to shocks. The scheme achieves high-order accuracy by coupling spatial reconstructions, such as ENO (essentially non-oscillatory) and WENO-AO (weighted essentially non-oscillatory with adaptive order), with strong stability-preserving Runge-Kutta methods for time discretisation. Dimensional splitting extends the method to higher-dimensional problems.\N\NThe authors validate their methodology through numerical experiments on benchmark problems, including Burgers' equation with shock formation and two-dimensional scalar conservation laws. The results demonstrate that the proposed scheme captures shocks and rarefaction waves with high fidelity and maintains stability and accuracy at large CFL numbers. The merging algorithm ensures that intersecting characteristics are effectively resolved, allowing the scheme to handle strong nonlinearities while retaining computational efficiency. A comparative analysis with existing methods reveals that the forward EL-RK-FV approach achieves superior performance in terms of error reduction and convergence.\N\NThis work holds significant implications for computational mathematics and related disciplines, as it presents a robust framework for tackling transport-dominated hyperbolic problems. By leveraging advanced reconstruction techniques and adaptive domain partitioning, the proposed method facilitates accurate and efficient simulations of complex systems. This contribution enhances the theoretical and practical understanding of numerical schemes for conservation laws, offering a versatile tool for researchers and practitioners in fields requiring high-precision modelling of dynamic systems.