Locally conservative and flux consistent iterative methods

TL;DR

This paper extends the concepts of conservation and flux consistency to iterative methods for hyperbolic conservation laws, analyzing their properties, convergence, and practical enforcement, with numerical validation on Euler equations.

Contribution

It introduces formal definitions of locally conservative and flux consistent iterative methods, extending the Lax-Wendroff theorem, and demonstrates how to enforce flux consistency and establish local conservation in various methods.

Findings

Pseudo-time iterations are locally conservative but not flux consistent.

An extended Lax-Wendroff theorem shows convergence towards weak solutions.

Numerical experiments validate theoretical results and case-dependent impact of flux consistency.

Abstract

Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the spatial flux terms. A technique for enforcing flux consistency, and thereby recovering convergence, is presented. Further, local conservation is established for all Krylov subspace methods, with and without restarts, and for Newton's method under certain assumptions on the discretization. Thus it is shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2D compressible Euler equations corroborate the theoretical results. Further numerical investigations of the impact of flux consistency on Newton-Krylov methods indicate that its effect is case dependent, and diminishes as the number of iterations grow.