Numerical methods for time-dependent convection-diffusion equations
Journal of Computational and Applied Mathematics
Numerical computation of internal & external flows: fundamentals of numerical discretization
Numerical computation of internal & external flows: fundamentals of numerical discretization
Applied Numerical Mathematics
A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems
Journal of Computational and Applied Mathematics
Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions
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A numerical approach is proposed to examine the singularly perturbed time-dependent convection-diffusion equation in one space dimension on a rectangular domain. The solution of the considered problem exhibits a boundary layer on the right side of the domain. We semi-discretize the continuous problem by means of the Crank-Nicolson finite difference method in the temporal direction. The semi-discretization yields a set of ordinary differential equations and the resulting set of ordinary differential equations is discretized by using a midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost second-order accurate in a coarse mesh and almost first-order accurate in a fine mesh in the spatial direction. The accuracy achieved in the temporal direction is almost second order. An extensive amount of analysis has been carried out in order to prove the uniform convergence of the method. Finally we have found that the resulting method is uniformly convergent with respect to the singular perturbation parameter, i.e. ε-uniform. Some numerical experiments have been carried out to validate the proposed theoretical results.