Numerical solution of the porous medium equation with source terms using the Four-Point Newton-EGKSOR iterative method combined with wave variable transformation

Abstract

The porous medium equation with source terms (PMES) is a nonlinear degenerate parabolic equation and serves as a model of many physical phenomena. To obtain its exact solution is a difficult task; hence it is necessary to find the approximate solution for the equation. In this paper, we proposed the 4-point Newton-Explicit Group Kaudd-SOR (4NEGKSOR) iterative method combined with the similarity transformation to solve the PMES numerically and obtain its approximate solution. The similarity transformation will be used to reduce the PMES into an ordinary differential equation, and we discretized the reduced form of the PMES using the finite difference scheme. Further, the processes for generating an approximation solution of the PMES proceeded via the 4NEGKSOR, and its formulation is derived. Moreover, the proposed method was tested with some numerical experiments to verify its effectiveness against existing iterative methods, i.e., the Newton-Gauss Seidal (NGS) and the Newton-Kaudd SOR (NKSOR). Based on the obtained results, the 4NEGKSOR iterative method proposed in this work is more efficient in getting the converged solution of the PMES compared to NGS and NKSOR iterative methods.