High order semi-implicit multistep methods for time dependent partial differential equations

2020/01/12 by Albi, Giacomo, Pareschi, Lorenzo
#35K57 #65L06 #65M12 #76Rxx #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2001.03974

Abstract

We consider the construction of semi-implicit linear multistep methods which 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, Filbet and Russo (2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allows, 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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