High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

Gottlieb, Sigal; Grant, Zachary J.; Hu, Jingwei; Shu, Ruiwen

doi:10.1137/21m1403175

High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

Journal Article · Mon Feb 14 23:00:00 EST 2022 · SIAM Journal on Numerical Analysis

DOI:https://doi.org/10.1137/21m1403175· OSTI ID:1867763

^[1]; ^[2]; Hu, Jingwei ^[3]; Shu, Ruiwen ^[4]

Univ. of Massachusetts, Dartmouth, MA (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Purdue Univ., West Lafayette, IN (United States)
Univ. of Maryland, College Park, MD (United States)

In this article we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.

View Accepted Manuscript (DOE)

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

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

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1867763

Journal Information:: SIAM Journal on Numerical Analysis, Journal Name: SIAM Journal on Numerical Analysis Journal Issue: 1 Vol. 60; ISSN 0036-1429

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
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