A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (EL–RK–FV) Method for Scalar Nonlinear Conservation Laws

Chen, Jiajie; Nakao, Joseph; Qiu, Jing-Mei; Yang, Yang

doi:10.1007/s10915-024-02714-y

A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (EL–RK–FV) Method for Scalar Nonlinear Conservation Laws

Journal Article · Mon Nov 18 00:00:00 EST 2024 · Journal of Scientific Computing

DOI:https://doi.org/10.1007/s10915-024-02714-y· OSTI ID:2477708

Chen, Jiajie; ; Qiu, Jing-Mei; Yang, Yang

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.

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Sponsoring Organization:: USDOE

Grant/Contract Number:: SC0023164

OSTI ID:: 2477708

Journal Information:: Journal of Scientific Computing, Journal Name: Journal of Scientific Computing Journal Issue: 1 Vol. 102; ISSN 0885-7474

Publisher:: Springer Science + Business MediaCopyright Statement

Country of Publication:: United States

Language:: English

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A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (EL–RK–FV) Method for Scalar Nonlinear Conservation Laws

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