Numerical Simulation that Provides Non-oscillatory Solutions for Porous Medium Equation and Error Approximation of Boussinesq's Equation

摘要

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 L_(1), L_(2) 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.