# High Order Explicit Local Time-Stepping Methods For Hyperbolic Conservation Laws

## TL;DR

This paper develops high order explicit local time-stepping methods for hyperbolic conservation laws, enabling variable time steps across regions while ensuring stability and conservation, demonstrated through numerical experiments.

## Contribution

It introduces a novel predictor-corrector LTS framework based on SSP-RK schemes for hyperbolic conservation laws, achieving up to fourth-order accuracy.

## Key findings

- Methods are proven to be conservative and stable for scalar laws.
- Numerical experiments show high accuracy and efficiency.
- Framework allows spatially variable time stepping with strong stability properties.

## 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 (RKDG) methods and design LTS algorithms based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integrations in different regions. 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 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.