A Hybrid Finite Difference Scheme with Shishkin Mesh for Solving Singularly Perturbed One-Dimensional Parabolic Problems
by
Mohan K Kadalbajoo
Department Of Mathematics & Statistics, I.I.T. Kanpur, Kanpur-208016,India
Coauthors: Vikas Gupta

In the present paper, a class of singularly perturbed one-dimensional convection-diffusion parabolic problems are studied on a rectangular domain in the x-t plane. The solution of this class of problems possesses a regular (or exponential) boundary layer in the neighborhood of right part of the lateral surface of the domain. Classical finite difference schemes on uniform mesh are known to be inadequate to solve such problems. To overcome the shortcomings, we construct a finite difference scheme that comprises of Crank-Nicolson method to discretize in temporal direction on uniform mesh and a special hybrid finite difference scheme to discretize the special variable with piecewise uniform mesh Shishkin mesh. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. These bounds are applied in the convergence analysis of the proposed hybrid finite difference scheme on Shishkin mesh. The method has been shown to be second order convergent in the temporal variable and almost second order accurate in the spetial variable It is also proved that the proposed method is uniformly convergent with respect to the singular perturbation parameter . Higher accuracy and convergence of the method is demonstrated by numerical examples and experimentation.

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Date received: March 9, 2008