Postprocessing technique of the discontinuous Galerkin method for solving delay differential equations

IF 2.4 3区 数学 Q1 MATHEMATICS Journal of Applied Mathematics and Computing Pub Date : 2024-05-10 DOI:10.1007/s12190-024-02114-3
Qunying Tu, Zhe Li, Lijun Yi
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Abstract

We introduce an innovative postprocessing technique aimed at refining the accuracy of the discontinuous Galerkin method for solving linear delay differential equations (DDEs) with vanishing delays. The fundamental idea behind this postprocessing technique is to enhance the discontinuous Galerkin solution of degree k by incorporating a generalized Jacobi polynomial of degree \(k+1\). We demonstrate that this postprocessing step enhances convergence by one order under the \(L^\infty \)-norm. Moreover, we apply this technique to both nonlinear DDEs with vanishing delays and linear DDEs with non-vanishing delays. We further validated the theoretical results through a series of numerical examples.

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解决延迟微分方程的非连续伽勒金方法的后处理技术
我们介绍了一种创新的后处理技术,旨在提高非连续 Galerkin 方法求解具有消失延迟的线性延迟微分方程(DDE)的精度。这种后处理技术的基本思想是通过加入一个广义的雅可比多项式(degree \(k+1\))来增强k度的非连续Galerkin解。我们证明,在 \(L^\infty \)-规范下,后处理步骤将收敛性提高了一个阶。此外,我们还将这一技术应用于延迟消失的非线性 DDE 和延迟不消失的线性 DDE。我们通过一系列数值实例进一步验证了理论结果。
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来源期刊
Journal of Applied Mathematics and Computing
Journal of Applied Mathematics and Computing Mathematics-Computational Mathematics
CiteScore
4.20
自引率
4.50%
发文量
131
期刊介绍: JAMC is a broad based journal covering all branches of computational or applied mathematics with special encouragement to researchers in theoretical computer science and mathematical computing. Major areas, such as numerical analysis, discrete optimization, linear and nonlinear programming, theory of computation, control theory, theory of algorithms, computational logic, applied combinatorics, coding theory, cryptograhics, fuzzy theory with applications, differential equations with applications are all included. A large variety of scientific problems also necessarily involve Algebra, Analysis, Geometry, Probability and Statistics and so on. The journal welcomes research papers in all branches of mathematics which have some bearing on the application to scientific problems, including papers in the areas of Actuarial Science, Mathematical Biology, Mathematical Economics and Finance.
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