High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Giacomo Albi; Lorenzo Pareschi

doi:10.1007/s42967-020-00110-5

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Giacomo Albi^*
, Lorenzo Pareschi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

44 Downloads (Pure)

Abstract

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

Original language	English
Pages (from-to)	701-718
Number of pages	18
Journal	Communications on Applied Mathematics and Computation
Volume	3
Issue number	4
DOIs	https://doi.org/10.1007/s42967-020-00110-5
Publication status	Published - Dec 2021

Keywords

High order accuracy
Implicit-explicit methods
Multistep methods
Semi-implicit methods
Strong stability preserving

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1007/s42967-020-00110-5Licence: CC BY

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Cite this

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abstract = "We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.",

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T1 - High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

AU - Albi, Giacomo

AU - Pareschi, Lorenzo

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N2 - We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

AB - We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

KW - High order accuracy

KW - Implicit-explicit methods

KW - Multistep methods

KW - Semi-implicit methods

KW - Strong stability preserving

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High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Abstract

Keywords

ASJC Scopus subject areas

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