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

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