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HIGH ORDER EXPLICIT SECOND DERIVATIVE METHODS WITH STRONG STABILITY PROPERTIES BASED ON TAYLOR SERIES CONDITIONS

Part of: Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  23 September 2022

A. MORADI Open the ORCID record for A. MORADI [Opens in a new window] ,
A. ABDI Open the ORCID record for A. ABDI [Opens in a new window]  and
G. HOJJATI Open the ORCID record for G. HOJJATI [Opens in a new window]
A. MORADI
Affiliation:
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran; e-mail: a_moradi@tabrizu.ac.ir, ghojjati@tabrizu.ac.ir.
A. ABDI*
Affiliation:
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran; e-mail: a_moradi@tabrizu.ac.ir, ghojjati@tabrizu.ac.ir. Research Department of Computational Algorithms and Mathematical Models, University of Tabriz, Tabriz, Iran.
G. HOJJATI
Affiliation:
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran; e-mail: a_moradi@tabrizu.ac.ir, ghojjati@tabrizu.ac.ir. Research Department of Computational Algorithms and Mathematical Models, University of Tabriz, Tabriz, Iran.
*
e-mail: a_abdi@tabrizu.ac.ir
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Abstract

When faced with the task of solving hyperbolic partial differential equations (PDEs), high order, strong stability-preserving (SSP) time integration methods are often needed to ensure preservation of the nonlinear strong stability properties of spatial discretizations. Among such methods, SSP second derivative time-stepping schemes have been recently introduced and used for evolving hyperbolic PDEs. In previous works, coupling of forward Euler and a second derivative formulation led to sufficient conditions for a second derivative general linear method (SGLM), which preserve the strong stability properties of spatial discretizations. However, for such methods, the types of spatial discretizations that can be used are limited. In this paper, we use a formulation based on forward Euler and Taylor series conditions to extend the SSP SGLM framework. We investigate the construction of SSP second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs with order $p=r=s$ and stage order $q=p,p-1$ up to order eight, where r is the number of external stages and s is the number of internal stages of the method. Proposed methods are examined on some one-dimensional linear and nonlinear systems to verify their theoretical order, and show potential of these schemes in preserving some nonlinear stability properties such as positivity and total variation.

Keywords

second derivative methodsmonotonicitystrong stability preservingTaylor seriesEuler equation

MSC classification

Primary: 65L05: Initial value problems
Type
Research Article
Information
The ANZIAM Journal , Volume 64 , Issue 3 , July 2022 , pp. 264 - 291
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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