Implicit and implicit--explicit high-order BDF methods for coupled elliptic--parabolic systems

TL;DR

This paper develops and analyzes high-order implicit and implicit-explicit BDF schemes for coupled elliptic-parabolic systems, achieving higher accuracy with minimal additional computational cost and establishing convergence under specific conditions.

Contribution

It introduces new high-order (up to sixth-order) implicit and implicit-explicit BDF schemes for coupled systems, with convergence analysis and improved accuracy.

Findings

Implicit-explicit schemes are decoupled, enhancing efficiency.

Fully implicit schemes require no coupling conditions.

Higher-order schemes significantly improve accuracy at similar computational costs.

Abstract

First-order fully implicit as well as implicit--explicit schemes for coupled elliptic-parabolic systems are discussed in [Ern and Meunier, ESAIM: M2AN, 2009] and [Altmann et al., Math.\ Comp., 2021], respectively. The extension of the analysis to higher-order (third-, fourth-, fifth-, and sixth-order) schemes is not straightforward since explicitly constructing $G$ matrices (G-stability) is often tricky. In this article, we develop fully implicit as well as implicit--explicit backward difference formula (BDF) schemes of order up to six. The implicit--explicit variants are decoupled, thereby enhancing computational efficiency; their convergence analysis requires a weak coupling condition on the poroelastic parameters. In contrast, no coupling conditions are needed for the fully implicit, coupled schemes. We determine novel and suitable multipliers for the two proposed classes and establish error estimates via the energy technique. A prominent advantage of these higher-order schemes is that, with almost the computational cost of first-order schemes, they greatly improve the accuracy.