Constraint-satisfying Krylov solvers for structure-preserving discretisations

TL;DR

This paper introduces a modified Krylov solver that enforces constraints to preserve geometric properties in structure-preserving discretisations of PDEs, enabling efficient solutions for large systems while maintaining key properties.

Contribution

It proposes a constraint-enforcing modification to the FGMRES iterative solver for structure-preserving discretisations of PDEs, improving practicality for large systems.

Findings

Enforces constraints on approximate solutions in Krylov methods.

Applicable to conservation laws and dissipative systems.

Enhances efficiency of structure-preserving schemes.

Abstract

A key consideration in the development of numerical schemes for time-dependent partial differential equations (PDEs) is the ability to preserve certain properties of the continuum solution, such as associated conservation laws or other geometric structures of the solution. There is a long history of the development and analysis of such structure-preserving discretisation schemes, including both proofs that standard schemes have structure-preserving properties and proposals for novel schemes that achieve both high-order accuracy and exact preservation of certain properties of the continuum differential equation. When coupled with implicit time-stepping methods, a major downside to these schemes is that their structure-preserving properties generally rely on exact solution of the (possibly nonlinear) systems of equations defining each time-step in the discrete scheme. For small systems, this is often possible (up to the accuracy of floating-point arithmetic), but it becomes impractical for the large linear systems that arise when considering typical discretisation of space-time PDEs. In this paper, we propose a modification to the standard flexible generalised minimum residual (FGMRES) iteration that enforces selected constraints on approximate numerical solutions. We demonstrate its application to both systems of conservation laws and dissipative systems.