High Order semi-implicit Rosenbrock type and Multistep methods for evolutionary partial differential equations with higher order derivatives

TL;DR

This paper develops semi-implicit Rosenbrock and multistep methods for high-order PDEs, enabling larger time steps without Newton iterations and demonstrating stability and accuracy through numerical experiments.

Contribution

It introduces flexible semi-implicit schemes for high-order PDEs that avoid severe time-step restrictions and do not require Newton iterations.

Findings

Schemes up to order 4 are constructed and tested.

Numerical experiments confirm stability and expected accuracy.

Methods effectively handle dissipative, dispersive, and biharmonic equations.

Abstract

The aim of this work is to apply a semi-implicit (SI) strategy within a Rosenbrock-type and IMEX linear multistep (LM) framework to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy provides great flexibility to treat these equations, and allows the construction of simple lienarly implicit schemes without any Newton iteration. Furthermore, the SI schemes so designed do not require the severe time-step restrictions typically encountered when using explicit methods for stability, i.e., $\Delta t = \mathcal{O}(\Delta x^k)$ for the $k$-th order PDEs with $k\ge 2$. For space discrertization, this strategy is combined with finite difference schemes. We provide example of methods up to order $p = 4$, and we illustrate the effectiveness of the schemes with appllications to dissipative, dispersive, and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and achieve the expected orders of accuracy.