High order explicit local time stepping methods for hyperbolic conservation laws
High order explicit local time stepping methods for hyperbolic conservation laws
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DOI：
10.1090/mcom/3507
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发表时间：
2019-05
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影响因子：
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通讯作者：
Thi-Thao-Phuong Hoang;L. Ju;W. Leng;Zhu Wang
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文献类型：
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作者：
Thi-Thao-Phuong Hoang;L. Ju;W. Leng;Zhu Wang
本文提出并分析了构造双曲守恒律的高阶显式局部时间步进（LTS）方法的一般框架。特别地，我们考虑了由龙格-库塔不连续伽辽金（RKDG）方法离散化的模型问题，并设计了基于强稳定性保持龙格-库塔（SSP-RK）格式的LTS算法，该算法允许在不同区域使用空间可变的时间步长进行时间积分。所提出的算法是一种预测校正型算法，首先基于SSP-RK近似和Taylor展开式对界面信息沿时间方向进行预测，然后对界面区域上的通量进行校正，使其在每个时间步长都能准确地保持质量。根据所提出的框架，详细介绍了精度高达四阶的LTS格式，并证明了它们的守恒性和标量守恒律的非线性稳定性。数值实验也证明了所提出的LTS算法的优异性能。
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.