The dispersion error is often the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of $\alpha$-dispersion-relation-preserving schemes. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which preserve the dispersion-relation of the continuous problem uniformly to an $\alpha \%$-error tolerance. We give a general framework to design provably stable finite difference operators that preserve the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ($\pi$-mode) present on any equidistant grid at a tolerance of $5\%$ error. This significantly improves on the current standard that have a tolerance of $100 \%$ error.
翻译:分散误差往往是计算高频元件波波波传播问题解决方案的主要错误。 在本文中, 我们定义并给出了 $\ alpha$- dismission- relation- preservation- prespective 计划的明确例子。 这些是双曲部分偏差方程系统的双面有限差异方案, 将持续问题的分散- relence 统一维持在 $\ alpha ⁇ ⁇ $- eror 容忍度上。 我们提供了一个总体框架, 用于设计可察觉的稳定有限差异运算器, 以维护弹性波等双向波等系统的分散关系。 我们在此生成的操作器可以解决位于任何等离子网中的最大频率 (\ pi$- mode), 以 5 $ 错误的容忍度为 。 这大大改善了当前标准, 容忍度为 $ 100 ⁇ $ 错误 的标准 。