A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

doi:10.1007/s42967-019-0001-3

摘要/Abstract

摘要： High-order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multi-derivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second-derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second-derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufcient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p=6, and defne an optimization procedure that allows us to fnd such SSP methods. Several types of these methods are presented and their efciency compared. Finally, these methods are tested on several PDEs to demonstrate the beneft of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

关键词: Strong stability preserving, Taylor series, Hyperbolic conservation laws, Two derivative Runge Kutta

Abstract: High-order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multi-derivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second-derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second-derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufcient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p=6, and defne an optimization procedure that allows us to fnd such SSP methods. Several types of these methods are presented and their efciency compared. Finally, these methods are tested on several PDEs to demonstrate the beneft of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

Key words: Strong stability preserving, Taylor series, Hyperbolic conservation laws, Two derivative Runge Kutta

中图分类号:

Zachary Grant, Sigal Gottlieb, David C. Seal. A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 21-59.

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