A posteriori mesh method for a system of singularly perturbed initial value problems

Zhongdi Cen; Jian Huang; Aimin Xu; Zhongdi Cen; Jian Huang; Aimin Xu

doi:10.3934/math.2022917

AIMS Mathematics

2022, Volume 7, Issue 9: 16719-16732. doi: 10.3934/math.2022917

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Research article

A posteriori mesh method for a system of singularly perturbed initial value problems

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Received: 09 March 2022 Revised: 25 June 2022 Accepted: 04 July 2022 Published: 12 July 2022
MSC : 65L10, 65L12

A system of singularly perturbed initial value problems with weak constrained conditions on the coefficients is considered. First the system of second-order singularly perturbed problems is transformed into a system of first-order singularly perturbed problems with integral terms, which facilitates the subsequent stability and a posteriori error analyses. Then a hybrid difference method with the use of interpolating quadrature rules is utilized to approximate the transformed system. Next a posteriori error analysis for the discretization scheme on an arbitrary mesh is presented. A solution-adaptive algorithm based on a posteriori error estimation is devised to generate a posteriori mesh and obtain approximation solution. Finally numerical experiments show a uniform convergence behavior of second-order for the scheme, which improves the previous results and achieves the optimal convergence order under the given discrete scheme.
- singular perturbation,
- hybrid difference scheme,
- mesh equidistribution,
- a posteriori error analysis,
- second order problems
Citation: Zhongdi Cen, Jian Huang, Aimin Xu. A posteriori mesh method for a system of singularly perturbed initial value problems[J]. AIMS Mathematics, 2022, 7(9): 16719-16732. doi: 10.3934/math.2022917

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Abstract

A system of singularly perturbed initial value problems with weak constrained conditions on the coefficients is considered. First the system of second-order singularly perturbed problems is transformed into a system of first-order singularly perturbed problems with integral terms, which facilitates the subsequent stability and a posteriori error analyses. Then a hybrid difference method with the use of interpolating quadrature rules is utilized to approximate the transformed system. Next a posteriori error analysis for the discretization scheme on an arbitrary mesh is presented. A solution-adaptive algorithm based on a posteriori error estimation is devised to generate a posteriori mesh and obtain approximation solution. Finally numerical experiments show a uniform convergence behavior of second-order for the scheme, which improves the previous results and achieves the optimal convergence order under the given discrete scheme.

References

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