An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type

Abstract

In this work we are interested in the numerical approximation of the solutions to 1D semilinear parabolic singularly perturbed systems of reaction-diffusion type, in the general case where the diffusion parameters for each equation can have different orders of magnitude. The numerical method combines the classical central finite differences scheme to discretize in space and a linearized fractional implicit Euler method together with a splitting by components technique to integrate in time. In this way, only tridiagonal linear systems must be solved to compute the numerical solution; consequently, the computational cost of the algorithm is considerably less than that of classical schemes. If the spatial discretization is defined on appropriate nonuniform meshes, the method is uniformly convergent of first order in time and almost second order in space. Numerical results for some test problems are presented which corroborate in practice the uniform convergence and the efficiency of the algorithm.

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Funding

This research was partially supported by the projects MTM2014-52859-P and MTM2017-83490-P and the Aragón Government and European Social Fund (group E24-17R).

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Correspondence to C. Clavero.

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Clavero, C., Jorge, J.C. An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type. Numer Algor 85, 1005–1027 (2020). https://doi.org/10.1007/s11075-019-00850-3

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Keywords

  • Semilinear parabolic systems
  • Linearly implicit methods
  • Splitting by components
  • Nonuniform meshes
  • Uniform convergence

Mathematics Subject Classification (2010)

  • 65N06
  • 65N12
  • 65M06