Partitioned and implicit-explicit general linear methods for ordinary differential equations

TL;DR

This paper introduces new implicit-explicit general linear methods for solving differential equations with stiff and nonstiff parts, providing a theoretical framework and practical schemes with confirmed high-order accuracy.

Contribution

It develops an order conditions theory for high stage order partitioned GLMs and constructs practical third-order IMEX schemes based on diagonally-implicit multi-stage methods.

Findings

Theoretical framework for high stage order partitioned GLMs

Construction of practical third-order IMEX schemes

Numerical results confirm theoretical accuracy

Abstract

Implicit-explicit (IMEX) time stepping methods can efficiently solve differential equa- tions with both stiff and nonstiff components. IMEX Runge-Kutta methods and IMEX linear multistep methods have been studied in the literature. In this pa- per we study new implicit-explicit methods of general linear type (IMEX-GLMs). We develop an order conditions theory for high stage order partitioned GLMs that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order three. Numerical results confirm the theoretical findings.