论文标题
非线性保护法的三阶MUSCL方案的截断误差分析
A Truncation Error Analysis of Third-Order MUSCL Scheme for Nonlinear Conservation Laws
论文作者
论文摘要
本文反驳了文献中发现的MUSCL方案不能对非线性保护法律准确的三阶准确性。我们基于仔细且详细的截断误差分析，为MUSCL方案的三阶准确性提供了严格的证明。在整个分析过程中，将严格针对数值解决方案以及目标操作员进行分区平均值与点值之间的区别。结果表明，Van Leer's Kappa-Scheme在面部重建的溶液的平均值准确地恢复了一个立方体溶液，而Kappa = 1/3 = 1/3，对于由重建的溶液评估的非线性通量的平均值也是如此，并且耗散项已经足够小，而耗散术语已经足够小。最后，指出目标空间操作员是一个细胞平均通量导数，我们证明MUSCL有限数量方案的前导截断误差是第三阶，kappa = 1/3。还讨论了扩散方案的重要性：当将三阶MUSLC方案与错误的四阶扩散方案一起用于对流扩散问题时，将丢失三阶精度。通过彻底的数值实验来验证三阶准确性，以解决稳定和不稳定问题。本文旨在作为澄清MUSCL方案三阶准确性的困惑的参考，以作为正确分析和验证非线性方程的MUSCL方案的指南，并最终作为阐明后续论文中阐明三阶非结构性网格方案的基础。
This paper is a rebuttal to the claim found in the literature that the MUSCL scheme cannot be third-order accurate for nonlinear conservation laws. We provide a rigorous proof for third-order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's kappa-scheme recovers a cubic solution exactly with kappa = 1/3, the same is true for the average of the nonlinear fluxes evaluated by the reconstructed solutions, and a dissipation term is already sufficiently small with a third-order truncation error. Finally, noting that the target spatial operator is a cell-averaged flux derivative, we prove that the leading truncation error of the MUSCL finite-volume scheme is third-order with kappa = 1/3. The importance of the diffusion scheme is also discussed: third-order accuracy will be lost when the third-order MUSLC scheme is used with a wrong fourth-order diffusion scheme for convection-diffusion problems. Third-order accuracy is verified by thorough numerical experiments for both steady and unsteady problems. This paper is intended to serve as a reference to clarify confusions about third-order accuracy of the MUSCL scheme, as a guide to correctly analyze and verify the MUSCL scheme for nonlinear equations, and eventually as the basis for clarifying third-order unstructured-grid schemes in a subsequent paper.
