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.

中文翻译：

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具有修正安德森加速度的稳态热传导方程保持最大值原理的有限体积法

摘要 在本文中，我们提出了一种有限体积方案，该方案保留了失真网格上稳态热传导方程的离散最大值原理 (DMP)。与这些保留 DMP 的有限体积方案相反，我们的新方案使用两个单侧数值热通量的几何平均值（而不是调和平均值），特别是它产生了更准确的通量近似值，这在数值上得到了验证。我们证明存在 DMP 和我们方案的解决方案的存在。我们还提出了一种改进的安德森加速（MAA）算法来提高鲁棒性并加速收敛。算法设计基于非线性系统残差向量线性组合的最小化问题。数值实验验证了我们方案的 DMP 保留特性和 MAA 算法的效率。此外，使用 MAA 的 Picard 迭代的稳定性和效率比使用经典 Anderson 加速的要好得多。在数值例子中，MAA迭代的收敛速度高达Picard迭代收敛速度的7倍。