Abstract
So far, so many works have been done in a different way to find the exact track that can grasp the true solution for the one-dimensional porous media equation (PME). For instance, Monika studied using relaxation [1], Q. Zhang and Z. Wu. have already done the similar work by local degenerate Galarkin (LDG) method [12] and so on. Still, this is a challenge to find an appropriate scheme that can track the true solution when adiabatic exponent increases monotonically. In this paper, we have studied numerical result for where we have used Explicit-Implicit Finite Difference Method (EIFDM). Since so far PME is a degenerate parabolic equation and analytically the existence and uniqueness occur weakly only in the Sobolev sense, it is very hard to track the true solution numerically. Our main objective is to study numerically the PME with mixed boundary conditions and shown the result is helpful to track the Barenblatt's self similar solution and its interface when adiabatic exponent larger than 3 that provides much less error. This paper will show a possibility that Finite Difference Method (FDM) is also helpful rather the Finite Elements Method to track the interface in the simulation with an appropriate initial guess. Also checked L1, L2 and L∞-error for Boussinesq's equation which is a fundamental equation of ground- water flow, hopefully, the simulated results can help when this equation is useful in the practical world. Finally, all studied results are given to show the advantage of the θ-scheme method in the simulation of the PME and its capability to capture accurately sharp interfaces without oscillation.
Highlights
The aim of this paper is to study numerical approximation of the nonlinear diffusion equation.ρt = ∆(ργ), γ > 1, (1)and usually it is called the porous medium equation (PME), with due attention paid to its closest relatives
Usually it is called the porous medium equation (PME), with due attention paid to its closest relatives
The equation can be posed for all x ∈ Rd and 0 < t < ∞, and initial conditions are needed to determine the solutions; but it is quite often posed, especially in practical problems, in a bounded sub-domain Ω ∈ Rd for 0 < t < T, and determination of a unique solution asks for boundary conditions as well as initial conditions.[4, 6]
Summary
The aim of this paper is to study numerical approximation of the nonlinear diffusion equation. The weak solution may lose its classical derivative at some (interface) points, and the sharp interface of support may propagate with finite speed if the initial data have compact support Enamored of these interesting facts, there have been many works on the simulation for the non smooth solution of the PME, for example, the finite different method by Graveleau and Jamet [25], the interface tracking algorithm by DiBenedetto and Hoff [26], and the relaxation scheme referred to in [1]. The second component is an analysis for the nonnegativity preservation principle of the considered θ-method, i.e., in each cell of numerical discretization of the Barenblatt solution [8] for the PME remains non-negative for different adiabatic exponents (see the Figure 7). To learn more about numerical methods for parabolic equations including PME, we invite the reader to take a closer look at one or several of the references [7, 9, 10, 11, 16, 17, 18, 20, 21, 24, 25, 26]
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