AI 中文总结
研究针对弗里德里希斯系统，构建高效高阶局部时间积分方案，核心方法是用预处理克雷洛夫子空间方法近似线性系统解，通过特定方式构建预处理器，经数值实验验证理论结果，实现高效积分。
AI 中文摘要
本文为空间离散化的线性弗里德里希斯系统构建了一种高效的高阶局部时间积分方案。特别关注只有少数网格单元小而多数单元大得多的问题。像Hochbruck和Sturm（2016年）以及Hochbruck和Köhler（2022年）那样在网格粗部分用蛙跳法、细部分用克兰克 - 尼科尔森法的特殊组合不适用于高阶时间积分。因此建议用预处理的克雷洛夫子空间方法近似每个时间步产生的线性系统的解，如弗罗因德和纳赫蒂加尔（1991年）的拟极小残差法。这里为线性问题开发的技术也适用于非线性问题。通过分析局部隐式方法，展示如何构建预处理器以使克雷洛夫子空间方法达到一定精度所需的迭代次数与小网格单元直径无关。利用法贝尔多项式和复逼近理论证明了这种行为。应用预处理器的成本包括求解一个小线性系统，其维数对应于网格细部分（及其下一个粗邻域）内自由度。若此维数比全网格尺寸小，则预处理器非常高效。最后用四阶高斯 - 勒让德龙格 - 库塔方法的数值实验验证了理论结果。
英文摘要
In this paper, we construct an efficient higher-order local time integration scheme for spatially discretized linear Friedrichs' systems. In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. The special combination of two methods like the leapfrog method on the coarse part of the mesh and the Crank-Nicolson method on the fine part as was done in Hochbruck, Sturm 2016 and Hochbruck, Köhler 2022 is not suitable for higher-order time integration. Therefore, we suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal 1991. The techniques developed here for linear problems also carry over to nonlinear problems, where linear systems of the same type arise within a Newton-type iteration. Motivated by the analysis of locally implicit methods by Hochbruck and Sturm 2016, we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory. The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient. We conclude by verifying our theoretical results with numerical experiments for the fourth-order Gauss-Legendre Runge--Kutta method.
