An Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for solving convection and convection-diffusion equations

• Published in: Journal of computational physics Vol. 470; p. 111589
• Main Authors: Nakao, Joseph, Chen, Jiajie, Qiu, Jing-Mei
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2022
• Subjects: Convection-diffusion equation; Eulerian-Lagrangian; IMEX Runge-Kutta; Semi-Lagrangian; WENO-AO; WENO-AO; Eulerian-Lagrangian; IMEX Runge-Kutta; Semi-Lagrangian; Convection-diffusion equation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111589

We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines approach. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory schemes with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit RK methods for convection equations and implicit-explicit (IMEX) RK methods for convection-diffusion equations. Our algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate our algorithm's robustness, high-order accuracy, and ability to handle extra large time steps. •The novel design of the EL RK FV framework that accommodates both the SL FV method and classical Eulerian RK FV method.•Partition of space-time region that follows dynamics of characteristics.•The time step constraint from an Eulerian method is greatly mitigated.