紧凑有限差分方案的分析与开发：优化数值色散关系
Analysis and development of compact finite difference schemes with optimized numerical dispersion relations
AI总结：
本文分析了优化数值色散关系的紧凑有限差分方案，探讨了色散与耗散误差的动态，并提出了一种优化方案以减少线性和非线性问题的误差。
摘要：
有限差分近似除了泰勒截断误差外，还会引入数值色散和耗散误差。本文分析了一类旨在最小化这些误差（以牺牲形式精度为代价）的有限差分方案，并研究了泰勒截断误差与色散-耗散误差在网格细化过程中的相互作用。特别地，我们研究了一维和二维非色散传输方程的完全离散数值色散关系。我们推导了解的$ L^2 $-范数误差与色散-耗散误差之间的关系。基于我们的分析，我们探讨了各种优化紧凑方案和非优化高阶广义Padé紧凑方案之间的误差动态。这些动态揭示了设计适合给定问题的优化紧凑方案的原则。使用这些原则作为指导，我们提出了一种优化方案，在确定离散化之前规定数值色散关系。这种方法在线性和非线性问题中产生了比非优化高阶紧凑方案和其他文献中开发的优化方案更小的数值色散和耗散误差。最后，我们讨论了开发具有非平凡边界条件问题的优化复合边界方案的难度，并提出了一种引入缓冲区以连接优化内部方案和非优化边界方案的复合方案。我们的数值实验表明，当波包通过非周期性边界时，这种策略会产生小的$ L^2 $-范数误差。
英文摘要：
Finite difference approximation, in addition to Taylor truncation errors, introduces numerical dispersion-and-dissipation errors into numerical solutions of partial differential equations. We analyze a class of finite difference schemes which are designed to minimize these errors (at the expense of formal order of accuracy), and we analyze the interplay between the Taylor truncation errors and the dispersion-and-dissipation errors during mesh refinement. In particular, we study the numerical dispersion relation of the fully discretized non-dispersive transport equation in one and two space dimensions. We derive the numerical phase error and the $L^2$-norm error of the solution in terms of the dispersion-and-dissipation error. Based on our analysis, we investigate the error dynamics among various optimized compact schemes and the unoptimized higher-order generalized Padé compact schemes. The dynamics shed light on the principles of designing suitable optimized compact schemes for a given problem. Using these principles as guidelines, we then propose an optimized scheme that prescribes the numerical dispersion relation before finding the corresponding discretization. This approach produces smaller numerical dispersion-and-dissipation errors for linear and nonlinear problems, compared with the unoptimized higher-order compact schemes and other optimized schemes developed in the literature. Finally, we discuss the difficulty of developing an optimized composite boundary scheme for problems with non-trivial boundary conditions. We propose a composite scheme that introduces a buffer zone to connect an optimized interior scheme and an unoptimized boundary scheme. Our numerical experiments show that this strategy produces small $L^2$-norm error when a wave packet passes through the non-periodic boundary.
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