Abstract
In this paper, a numerical method has been proposed for solving several two-dimensional porous medium equations (2D PME). The method combines Newton and Explicit Group MSOR (EGMSOR) iterative method namely four-point NEGMSOR. Throughout this paper, an initialboundary value problem of 2D PME is discretized by using the implicit finite difference scheme in order to form a nonlinear approximation equation. By taking a fixed number of grid points in a solution domain, the formulated nonlinear approximation equation produces a large nonlinear system which is solved using the Newton iterative method. The solution vector of the sparse linearized system is then computed iteratively by the application of the four-point EGMSOR method. For the numerical experiments, three examples of 2D PME are used to illustrate the efficiency of the NEGMSOR. The numerical result reveals that the NEGMSOR has a better efficiency in terms of number of iterations, computation time and maximum absolute error compared to the tested NGS, NEG and NEGSOR iterative methods. The stability analysis of the implicit finite difference scheme used on 2D PME is also provided.
Highlights
This paper considers the numerical solution of the following initial-boundary value problem of the two-dimensional porous medium equation (2D PME): u um u um u, (1)
There are several iterative methods that have been proposed in order to reduce the computational complexity and this paper considers the application of the Explicit Group (EG) method which was introduced by Evans [8]
This paper proposed the unconditionally stable implicit difference scheme together with the NEGMSOR iterative method for solving several examples of 2D PME
Summary
This paper considers the numerical solution of the following initial-boundary value problem of the two-dimensional porous medium equation (2D PME): u um u um u , (1). T x xy y that is subjected to the initial condition:. Equation (1) can be known as the generalized form of the heat equation. By referring to equation (1), the heat equation can be obtained by taking m 1. Equation (1) is a type of partial differential equation in the nonlinear degenerate parabolic class which is widely used to describe models for gases in porous media, heat diffusion, underground infiltration, population dynamics as well as high energy physics [1]. The presence of a nonlinear term u m in equation (1) makes the exact solution difficult to be obtained
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