Splitting schemes for coupled differential equations: Block Schur-based approaches & Partial Jacobi approximation

Nuca, Roberto; Storvik, Erlend; Radu, Florin A.; Icardi, Matteo

Abstract

Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. In a broad sense, one can categorise the numerical solution strategies for coupled problems into two classes: monolithic approaches and sequential (also known as split, decoupled, partitioned or segregated) approaches. The monolithic approaches treat the entire problem as one, whereas the sequential approaches are iterative decoupling techniques where the different sub-problems are treated separately. Although the monolithic approaches often offer the most robust solution strategies, they tend to require ad-hoc preconditioners and numerical implementations. Sequential methods, on the other hand, offer the possibility to add and remove equations from the model flexibly and rely on existing black-box solvers for each specific equation. Furthermore, when problems are non-linear, inner iterations need to be performed even in monolithic solvers, making the sequential approaches an even more viable alternative. The cost of running inner iterations to recover the multi-physics coupling could, however, easily become prohibitive. Moreover, the sequential approaches might not converge at all. In this work, we present a general formulation of splitting schemes for continuous operators with arbitrary implicit/explicit splitting, like in standard iterative methods for linear systems. By introducing a generic relaxation operator, we find the conditions for the convergence of the iterative schemes. We show how the relaxation operator can be thought of as a preconditioner and constructed based on an approximate Schur complement. We propose a Schur-based Partial Jacobi relaxation operator to stabilise the coupling and show its effectiveness. Although we mainly focus on scalar-scalar linear problems, most results are easily extended to non-linear and higher-dimensional problems. The schemes presented are not explicitly dependent on any particular discretisation methodologies. Numerical tests (1D and 2D) for two PDE systems, namely the Dual-Porosity model and a Quad-Laplacian operator, are carried out to investigate the practical implications of the theoretical results.

Citation

Nuca, R., Storvik, E., Radu, F. A., & Icardi, M. (2024). Splitting schemes for coupled differential equations: Block Schur-based approaches & Partial Jacobi approximation. Computers and Mathematics with Applications, 161, 190-201. https://doi.org/10.1016/j.camwa.2024.02.042

• Journal Article Type: Article
• Acceptance Date: Feb 24, 2024
• Online Publication Date: Mar 15, 2024
• Publication Date: May 1, 2024
• Deposit Date: Mar 18, 2024
• Publicly Available Date: Mar 26, 2024
• Journal: Computers & Mathematics with Applications
• Print ISSN: 0898-1221
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 161
• Pages: 190-201
• DOI: https://doi.org/10.1016/j.camwa.2024.02.042
• Keywords: Partitioned multi-physics; Splitting schemes; Approximate Schur complement; Partial Jacobi; Block-iterative schemes; Sequential coupling
• Public URL: https://nottingham-repository.worktribe.com/output/32470993
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0898122124000865?via%3Dihub
• Additional Information: This article is maintained by: Elsevier; Article Title: Splitting schemes for coupled differential equations: Block Schur-based approaches & Partial Jacobi approximation; Journal Title: Computers & Mathematics with Applications; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.camwa.2024.02.042; Content Type: article; Copyright: © 2024 The Author(s). Published by Elsevier Ltd.