Parameter-uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction–diffusion equations with boundary and interior layers

Math Methods in App Sciences

2021

We present a finite difference method for a system of two singularly perturbed initial-boundary value semilinear reaction-diffusion equations. The highest order derivatives are multiplied by small perturbation parameters of different magnitudes. The problem is discretized using a central difference scheme in space and backward difference scheme in time on a Shishkin mesh. The convergence analysis has been given, and it has been established that the method enjoys almost second-order parameter-uniform convergence in space and first-order in time. Numerical experiments are conducted to demonstrate the efficiency of the method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

Math Methods in App Sciences

2021

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“…Suitable conditions on the appropriate mesh generating functions are derived, which are sufficient for the parameter-uniform convergence of the method in the discrete maximum norm with an optimal error bound on the Bakhalov-Shishkin mesh and Shishkin mesh. Singularly perturbed system of steady and unsteady reaction-diffusion problems with continuous/discontinuous source terms are considered in [21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis for a class of coupled system of singularly perturbed time‐dependent convection‐diffusion equations with a discontinuous source term

Chawla

Numerical Methods Partial

2021

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In this article, we present a parameters‐uniform numerical method for a time‐dependent weakly coupled system of two convection‐diffusion equations that has a discontinuity, along the line x=d, in the source term. The second order term of each equation is multiplied by a small singular perturbation parameter and these parameters are assumed to be different in magnitude. On an appropriate Shishkin mesh, the considered problem is discretized using an upwind central difference scheme away from the line of discontinuity, and a particular upwind central difference scheme along the line of discontinuity. The numerical approximations produced by this scheme are uniformly convergent of first‐order in time and almost first‐order in space with respect to both perturbation parameters. Numerical results are presented to validate the theoretical results.

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Convergence Analysis of a Layer Resolving Numerical Techniquefor a Class of Coupled System of Singularly Perturbed Parabolic Convection-Diffusion Equations Having an Interface

Springer Proceedings in Mathematics &Amp; Statistics

2023