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A high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution

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  • Kumar, Sunil
  • Sumit,
  • Vigo-Aguiar, Jesus

Abstract

The purpose of this paper is to introduce a high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution. The discretization is based on the backward Euler scheme in time and a high order non-monotone scheme in space. In time direction we consider a uniform mesh, while in spatial direction we construct an adaptive mesh through equidistribution of a monitor function involving appropriate power of the solution’s second derivative. The method is analysed in two steps, splitting the time and space discretization errors. We establish that the method is uniformly convergent with optimal order having order one in time and order four in space. Further, we use the Richardson extrapolation technique for improving the order of convergence from one to two in time. Numerical experiments are presented to confirm the theoretically proven convergence result.

Suggested Citation

  • Kumar, Sunil & Sumit, & Vigo-Aguiar, Jesus, 2022. "A high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 287-306.
    • Handle: RePEc:eee:matcom:v:199:y:2022:i:c:p:287-306
    DOI: 10.1016/j.matcom.2022.03.025
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    References listed on IDEAS

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    1. Daba, Imiru Takele & Duressa, Gemechis File, 2022. "Collocation method using artificial viscosity for time dependent singularly perturbed differential–difference equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 201-220.
    2. Munyakazi, Justin B. & Patidar, Kailash C. & Sayi, Mbani T., 2019. "A robust fitted operator finite difference method for singularly perturbed problems whose solution has an interior layer," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 160(C), pages 155-167.
    3. Kumar, Sunil & Sumit, & Ramos, Higinio, 2021. "Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    4. Kabeto, Masho Jima & Duressa, Gemechis File, 2021. "Robust numerical method for singularly perturbed semilinear parabolic differential difference equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 537-547.
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