Numerical Approximation to Porous Medium Equation Using a Quarter-Sweep Based Finite Difference and Explicit Four-Points Group

Abstract

In this study, numerical approximation is used to tackle the one-dimensional nonlinear porous media problem. The emphasis is on using a quarter-sweep finite difference technique to apply the approximation. The quarter-sweep computation approach is highly helpful in terms of reducing complexity while solving big and sparse linear equations. This is done in order to produce a finite difference scheme that works well for the porous medium equation. Internal calculations are carried out using an explicit four-points group methodology, and the resulting nonlinear system is solved using the Newton method. To demonstrate the efficiency of the suggested finite difference approximation and the explicit four-points group technique together, several examples involving the porous medium equation are described. Analyzing the maximum number enables an assessment of the computational time and iteration efficiency of the numerical approach. The method's accuracy is further evaluated by looking at the largest absolute errors the grid points can produce. The suggested method is compared with a half-sweep finite difference method that uses the Newton explicit group method and a conventional quarter-sweep finite difference approximation.