Conservative iterative methods for implicit discretizations of conservation laws

TL;DR

This paper analyzes the conservation properties of various iterative methods applied to implicit discretizations of nonlinear conservation laws, revealing conditions for conservation, convergence, and consistency improvements.

Contribution

It demonstrates that many iterative methods are globally conservative and extends the Lax-Wendroff theorem to these methods, proposing strategies to ensure consistency.

Findings

Newton's method, Krylov methods, and pseudo-time iterations are globally conservative.

Explicit Runge-Kutta pseudo-time iterations can be locally conservative but may have inconsistent flux.

A strategy is proposed to ensure the numerical flux matches the conservation law, improving convergence.

Abstract

Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton's method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the corresponding numerical flux can be inconsistent with the conservation law. We prove an extension of the Lax-Wendroff theorem, which reveals that numerical solutions based on these methods converge to weak solutions of a modified conservation law where the flux function is multiplied by a particular constant. This constant depends on the choice of Runge-Kutta method but is independent of both the conservation law and the discretization. Consistency is maintained by ensuring that this constant equals unity and a strategy for achieving this is presented. Experiments show that this strategy improves the convergence rate of the pseudo-time iterations.