A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

In this paper, we produce ϵ , μ − uniform numerical method for a singularly perturbed parabolic differential equation with two parameters. To approximate the solution, we consider the implicit Euler method for time direction, the finite difference method for spatial direction on a uniform mesh, and the fitted operator method with multiple fitting factors. To accelerate the convergence of the method, the Richardson extrapolation method is applied. The results show that the proposed method is second-order convergent in both temporal and spatial directions. The convergence of the scheme is insensitive to the two perturbation parameters. Two model examples are considered to validate the applicability of the proposed method and produced more accurate results compared to some methods that appear in the literature. Matlab software is used to manipulate the results.

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[Retracted] A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model

Ali

Jaber

Yaseen

et al. 2022

Complexity

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In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.

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A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

Cited by 3 publications

References 21 publications

A computational method for the singularly perturbed delay pseudo-parabolic differential equations on adaptive mesh

A computational method for the singularly perturbed delay pseudo-parabolic differential equations on adaptive mesh

Fitted Operator Method Using Multiple Fitting Factors for Two Parameters Singularly Perturbed Parabolic Problems

[Retracted] A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model

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