High order explicit local time stepping methods for hyperbolic conservation laws

Hoang, Thi-Thao-Phuong; Ju, Lili; Leng, Wei; Wang, Zhu

doi:10.1090/mcom/3507

High order explicit local time stepping methods for hyperbolic conservation laws
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by Thi-Thao-Phuong Hoang, Lili Ju, Wei Leng and Zhu Wang;

Math. Comp. 89 (2020), 1807-1842

DOI: https://doi.org/10.1090/mcom/3507

Published electronically: January 29, 2020

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Abstract:

In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RK-DG) methods and design LTS algorithms based on the strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integration in different regions of the computational domain. The proposed algorithms are of predictor-corrector-type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of the interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we detail the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical experiments are also presented to demonstrate excellent performance of the proposed LTS algorithms.

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