Abstract
In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.
Highlights
We consider the singularly perturbed parabolic problem posed on the domain G Ωx × Ωt (0, 1) ×
Afterwards, Vulanovic and Nhan in [16] improved on what had already been done and proposed a new uniformly convergent numerical scheme. Both the methods proposed in [15, 17] have been analyzed on a piecewise-uniform Shishkin mesh and were proved to be almost third-order accuracy in space
We describe the uniform mesh for the temporal discretization of the domain Ωt and the Shishkin mesh for the spatial discretization of the domain Ωx
Summary
We consider the singularly perturbed parabolic problem posed on the domain G Ωx × Ωt (0, 1) × Afterwards, Vulanovic and Nhan in [16] improved on what had already been done and proposed a new uniformly convergent numerical scheme Both the methods proposed in [15, 17] have been analyzed on a piecewise-uniform Shishkin mesh and were proved to be almost third-order accuracy in space. Discretize in the spatial direction by utilizing a hybrid finite difference scheme on the Shishkin mesh. E ultimate goal of numerical methods for problem (1) is to obtain a series of discrete solutions so as to achieve a numerical approximation of the continuous solution Such that its error converges to 0 uniformly as N ⟶ + ∞, where N is the number of discretization on the spatial mesh.
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