A strong stability preserving analysis for explicit multistage two-derivative time-stepping schemes based on Taylor series conditions. 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 sufficient 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 define an optimization procedure that allows us to find such SSP methods. Several types of these methods are presented and their efficiency compared. Finally, these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.
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References in zbMATH (referenced in 7 articles , 1 standard article )
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- Ji, Xing; Zhao, Fengxiang; Shyy, Wei; Xu, Kun: A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh (2020)
- Moradi, A.; Abdi, A.; Farzi, J.: Strong stability preserving second derivative diagonally implicit multistage integration methods (2020)
- Moradi, Afsaneh; Abdi, Ali; Farzi, Javad: Strong stability preserving second derivative general linear methods with Runge-Kutta stability (2020)
- Grant, Zachary; Gottlieb, Sigal; Seal, David C.: A strong stability preserving analysis for explicit multistage two-derivative time-stepping schemes based on Taylor series conditions (2019)
- Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor two-step Runge-Kutta methods (2019)
- Moradi, Afsaneh; Farzi, Javad; Abdi, Ali: Strong stability preserving second derivative general linear methods (2019)
- Balsara, Dinshaw S.; Li, Jiequan; Montecinos, Gino I.: An efficient, second order accurate, universal generalized Riemann problem solver based on the HLLI Riemann solver (2018)