A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

Gottlieb, Sigal; Grant, Zachary J.

doi:10.1016/j.apm.2025.116270

A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

Journal Article · Sat Jul 05 00:00:00 EDT 2025 · Applied Mathematical Modelling

DOI:https://doi.org/10.1016/j.apm.2025.116270· OSTI ID:2571208

^[1]; Grant, Zachary J. ^[1]

University of Massachusetts, Dartmouth, MA (United States)

High order strong stability preserving time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs in a wide variety of application areas including fluid dynamics, magnetohydrodynamics, semiconductor devices, electromagnetics, and astrophysics. Over the past decade multiderivative time-stepping have been increasingly used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become important. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the strong stability preserving theory for explicit and implicit two-derivative Runge–Kutta schemes, including a special condition on the second derivative under which these implicit methods may be unconditionally strong stability preserving. This special condition is natural for the stiff component of wide range of plasma physics problems, and can be useful in the context of strong stability preserving implicit-explicit multi-derivative Runge–Kutta schemes, where the time-step restriction is then independent of the stiff term. Lastly, we present the strong stability preserving theory for implicit-explicit multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.

View Accepted Manuscript (DOE)

Research Organization:: University of Massachusetts, Dartmouth, MA (United States)

Sponsoring Organization:: Air Force Office of Scientific Research (AFOSR); National Science Foundation (NSF); USDOE; Office of Naval Research (ONR)

Grant/Contract Number:: SC0023164

OSTI ID:: 2571208

Journal Information:: Applied Mathematical Modelling, Journal Name: Applied Mathematical Modelling Vol. 149; ISSN 0307-904X

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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