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High order explicit local time stepping methods for hyperbolic conservation laws. (English) Zbl 1436.65130

Summary: 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.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35L65 Hyperbolic conservation laws

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