A finite volume method preserving maximum principle for steady heat conduction equations with modified Anderson acceleration

Abstract

In this article, we propose a finite volume scheme preserving the discrete maximum principle (DMP) for steady heat conduction equations on distorted meshes. In contrary to these finite volume schemes preserving DMP, our new scheme uses the geometric average (instead of harmonic average) of two one-side numerical heat fluxes, especially it produces a more accurate flux approximation, which is verified numerically. We prove that there hold the DMP and the existence of a solution for our scheme. We also propose a modified Anderson acceleration (MAA) algorithm to improve the robustness and accelerate the convergence. The algorithm design is based on a minimization problem for a linear combination of the residual vectors of the nonlinear system. Numerical experiments verify the DMP-preserving property of our scheme and the efficiency of the MAA algorithm. Moreover, the stability and efficiency of the Picard iteration with MAA are much better than that with the classical Anderson acceleration. In the numerical examples, the convergence rate of MAA iteration is up to seven times of the convergence rate of the Picard iteration.