非线性对流扩散方程的连续Petrov-Galerkin方法的数值研究

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Mathematical Analysis and Applications Pub Date : 2025-04-24 DOI:10.1016/j.jmaa.2025.129617

Zhihui Zhao, Hong Li, Wei Gao

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引用次数: 0

摘要

本文旨在利用连续Petrov-Galerkin （CPG）方法研究非线性对流扩散方程。该方法与有限元法同时对时间和空间变量进行离散化，便于在时间和空间上获得较高的阶精度，并具有较好的数值稳定性。此外，采用Petrov-Galerkin方法对模型问题进行近似，与常用的Galerkin方法相比，减少了计算规模。我们证明了CPG解的存在唯一性，并给出了不受空间网格参数约束的收敛性分析。通过数值试验验证了CPG方法的有效性和数值稳定性。数值试验结果表明，CPG法在求解非线性对流扩散方程方面优于标准有限元法和连续伽辽金法。

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The numerical study of a continuous Petrov-Galerkin method for the nonlinear convection-diffusion equation

This paper aims to use the continuous Petrov-Galerkin (CPG) method to study the nonlinear convection-diffusion equation. This method discretizes the time and space variables simultaneously with the finite element (FE) method, thus it is convenient to derive high order accuracy in time and space and has better numerical stability. In addition, the Petrov-Galerkin method is employed to approximate the model problem, which can reduce the computational scale in comparison with the usual Galerkin method. We demonstrate the existence and uniqueness of the CPG solution and give the convergence analysis without the constraints of spatial grid parameter. Several numerical tests are performed to access the validity and the numerical stability of the CPG method. Also, numerical tests illustrate that the CPG method is superior to the standard finite element (SFE) method and the continuous Galerkin (CG) method in solving the nonlinear convection-diffusion equation.

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来源期刊

Journal of Mathematical Analysis and Applications 数学-数学

CiteScore

2.50

自引率

7.70%

发文量

790

审稿时长

6 months

期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.