Locally Conservative and Flux Consistent Iterative Methods

Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax-Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are used to prove an extension of the Lax-Wendroff theorem incorporating pseudotime iterations with explicit Runge-Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of conservation laws, where each equation is modified by a particular scalar factor multiplying the spatial flux terms. Local conservation is further established for Krylov subspace methods with and without restarts, and for Newton's method under certain assumptions on the discretization. It is thus shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2 dimensional compressible Euler equations corroborate the theoretical results. A simple technique for enforcing flux consistency of Newton-Krylov methods is presented. Experiments indicate that its efficacy is case dependent, and diminishes as the number of iterations grow.