Multilevel Parareal Algorithm with Averaging for Oscillatory Problems

IF 3 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Scientific Computing Pub Date : 2024-08-20 DOI:10.1137/23m1547123

Juliane Rosemeier, Terry Haut, Beth Wingate

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引用次数: 0

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2709-A2736, August 2024.
Abstract. The present study is an extension of the work done by Peddle, Haut, and Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476–A3497] and Haut and Wingate [SIAM J. Sci. Comput., 36 (2014), pp. A693–A713], where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples.

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针对振荡问题的多级 Parareal 平均算法

SIAM 科学计算期刊》，第 46 卷第 4 期，第 A2709-A2736 页，2024 年 8 月。摘要本研究是对 Peddle、Haut 和 Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476-A3497] 以及 Haut 和 Wingate [SIAM J. Sci. Comput., 36 (2014), pp.本文提出的方法是一种具有任意多层次的多层次 Parareal 方法，它并不局限于两层情况。我们给出了一个渐近误差估计值，在只考虑两级的情况下，该估计值与两级估计值相减。引入两个以上的水平对平均过程有重要影响，因为我们为每个不同的水平选择了不同的平均窗口，这是本研究的另一个新特点。不同的平均窗口使所提出的方法特别适用于非线性多尺度问题，因为我们可以为问题的每个内在尺度引入一个层次，并调整平均程序，从而重现模型在该层次所解析的特定尺度上的行为。该方法适用于非线性微分方程。非线性会在问题中产生一系列频率。新方法的计算成本在几个例子中进行了调查和研究。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Scientific Computing 数学-应用数学

CiteScore

5.50

自引率

3.20%

发文量

209

审稿时长

1 months

期刊介绍： The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems. SISC papers are classified into three categories: 1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms. 2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist. 3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.