Variable-time-step weighted IMEX FEMs for nonlinear evolution equations

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED Applied Numerical Mathematics Pub Date : 2025-05-01 Epub Date: 2025-01-15 DOI:10.1016/j.apnum.2025.01.005

Meng Li , Dan Wang , Junjun Wang , Xiaolong Zhao

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Abstract

In this paper, the variable-time-step weighted implicit-explicit (IMEX) finite element methods (FEMs) are developed for some types of nonlinear real- or complex-valued evolution equations. Extensive research is conducted on the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels of the variable-time-step weighted IMEX scheme, elucidating their crucial properties in both real- and complex-valued scenarios. We prove that the scheme exhibits optimal convergence without any restrictions on the time-space step ratio. At last, several numerical examples are provided to demonstrate our theoretical results. With the weighted parameter

θ = 1

, the scheme in this work can degenerate into a special case: variable-time-step two-step backward differentiation formula (BDF2) scheme, and the convergence analysis in this special case was introduced in Liao et al. (2020) [15] and Liao et al. (2021) [29].

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非线性演化方程的变时间步长加权IMEX有限元

本文针对一类非线性实值或复值演化方程，建立了变时间步长加权隐显有限元方法。对变时间步长加权IMEX格式的离散正交卷积（DOC）核和离散互补卷积（DCC）核进行了广泛的研究，阐明了它们在实值和复值情况下的关键性质。证明了该方案在不受时空步长比限制的情况下具有最优收敛性。最后，通过数值算例对理论结果进行了验证。在加权参数θ=1的情况下，本文方案可以退化为一种特殊情况：变时间步两步后向微分公式（BDF2）方案，并在Liao et al.(2020)[15]和Liao et al.(2021)[29]中介绍了这种特殊情况下的收敛性分析。

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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.