A reduced-order two-grid method based on POD technique for the semilinear parabolic equation

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED Applied Numerical Mathematics Pub Date : 2024-07-29 DOI:10.1016/j.apnum.2024.07.012

Junpeng Song, Hongxing Rui

引用次数: 0

Abstract

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis.

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基于 POD 技术的半线性抛物方程降阶双网格法

在传统的双网格（TG）方法中，细网格上的非线性系统被转化为粗网格上的非线性子系统和细网格上的线性子系统，以降低计算成本。这种方法已成功应用于多个领域。然而，其计算效率仍然相对较低。为此，我们开发了一种新颖的减少自由度的双网格（ROTG）方法，用于求解半线性抛物方程。对于上述两个子系统，我们利用适当的正交分解（POD）技术来大幅减少自由度。我们还得出了 ROTG 方案的先验误差估计值。最后，我们进行了几次数值测试，以观察 ROTG 方法的行为并验证理论分析。

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

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.

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