Validated integration of semilinear parabolic PDEs

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Jan Bouwe van den Berg, Maxime Breden, Ray Sheombarsing
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Abstract

Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variation of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton–Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift–Hohenberg equation, the Ohta–Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.

Abstract Image

半线性抛物线 PDE 的验证整合
对进化偏微分方程(PDEs)进行积分是研究解的动态性的一个基本要素。事实上,模拟是科学计算的核心,但其数学可靠性往往难以量化,特别是当人们感兴趣的是给定模拟的输出,而不是离散化参数趋于零的渐近机制时。在本文中,我们提出了一种计算机辅助证明方法,用于对具有周期性边界条件的标量半线性抛物 PDE 进行严格的时间积分。我们根据傅里叶空间中的常数变化公式,提出了一个等效的寻零问题。利用切比雪夫插值法和域分解法,我们用牛顿-康托洛维奇类型的论证完成了证明。这一过程的最终结果是证明轨道的存在性,并保证该轨道与数值计算近似值之间的误差范围。我们用费雪方程、斯威夫特-霍恩伯格方程、Ohta-川崎方程和 Kuramoto-Sivashinsky 方程的结果来说明这种方法的多功能性。我们希望这种严格的积分器能成为研究偏微分方程中连接轨道的边界值问题的基础。
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
Numerische Mathematik
Numerische Mathematik 数学-应用数学
CiteScore
4.10
自引率
4.80%
发文量
72
审稿时长
6-12 weeks
期刊介绍: Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers: 1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis) 2. Optimization and Control Theory 3. Mathematical Modeling 4. The mathematical aspects of Scientific Computing
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