On quarter-sweep finite difference scheme for one-dimensional porous medium equations

In this article, we introduce an implicit finite difference approx-imation for one-dimensional porous medium equations using Quarter-Sweep approach. We approximate the solutions of the nonlinear porous medium equa-tions by the application of the Newton method and use the Gauss-Seidel itera-tion. This yields a numerical method that reduces the computational complex-ity when the spatial grid spaces are reduced. The numerical result shows that the proposed method has a smaller number of iterations, a shorter computation time and a good accuracy compared to Newton-Gauss-Seidel and Half-Sweep Newton-Gauss-Seidel methods

Implicit finite difference solution of 1D nonlinear porous medium equation via four-point EGSOR with newton iteration

Porous medium equation (PME) has a great practical in fluid flow, heat transfer and population dynamics. The nonlinearity in this equation makes it interesting in the development of nonlinear analytical and numerical tools in pure and applied mathematics and sciences. This paper proposes a Four-Point EGSOR with Newton iteration to solve the 1D PME problems. The reliability of proposed method is illustrated. The formulation and implementation of the proposed method are also presented. The numerical results showed that the Four-Point EGSOR with Newton iteration requires less number of iterations and computational time in obtaining the numerical solution to the 1D PME problems. With these results, it can be said that the Four-Point Newton-EGSOR iterative method can be a promising numerical method in tackling nonlinear differential equation problems. To enhance the rate of convergence of the current method, in future work, this study will investigate the application of MSOR as in Sulaiman et al. (2012)

Newton-SOR iterative method for solving the two-dimensional porous medium equation

In this paper, we consider the application of the Newton the approximate solution of the two nonlinear finite difference approximation equation to implicit finite difference scheme. The developed nonlinear system is linearized by using the Newton method. At each temporal step, the corresponding linear systems are solved by using SOR iteration. We investigate the eff three examples of 2D PME and the performance is compared with the Newton method. Numerical results show that the Newton Newton-GS iterative method in terms of a number of iterations, computer time and maximum absolute errors

Half-sweep newton-gauss-Seidel for implicit finite difference solution of 1d nonlinear porous medium equations

This paper proposes a new numerical technique called Half-Sweep Newton-Gauss-Seidel (HSNGS) iterative method in solvingone-dimensional nonlinear porous medium equations. The general form of porous medium equation (PME) is discretized by using implicit finite difference scheme which leads to a nonlinear finite difference approximation equation. The developed system of nonlinear equations is transformed by the application of Newton method into the corresponding system of linear equations. The numerical solutions are obtained by HSNGS iteration. Four illustrative examples are chosen in order to show the effectiveness of the proposed technique. The numerical results are compared with the Full-Sweep Newton-Gauss-Seidel (FSNGS) to demonstrate the applicability of the proposed iterative method. The HSNGS iterative method shows superiority in term of iteration number and computational time

Application of newton and MSOR methods for solving 2D porous medium equations

Nonlinear partial differential equations, for instance, porous medium equations, can be difficult to be solved. In the certain degree when the exact solution of a particular nonlinear differential equation is unworkable, the numerical approach can be the tool for an efficient solver. The numerical solution is important for a further investigation, not only in developing a better numerical method but also in studying the related complex phenomena. This paper aims to propose a numerical method that combines Newton and MSOR (NMSOR) methods for the solution of two-dimensional porous medium equations (2D PME). Implicit finite difference scheme is used to discretize the main nonlinear differential equation to generate a system of nonlinear equations. The system of nonlinear equations is then solved using the NMSOR method. The efficiency of the method in solving the nonlinear system is studied along with the tested Newton- Gauss-Seidel (NGS) and Newton-SOR (NSOR) methods. The finding shows the superiority of the NMSOR method over the other two tested methods in terms of a total number of iterations and time taken to obtain the solution. All three methods have a good agreement in terms of accuracy

Application of four-point newton- EGSOR iteration for the numerical solution of 2d porous medium equations

Partial differential equations that are used in describing the nonlinear heat and mass transfer phenomena are difficult to be solved. For the case where the exact solution is difficult to be obtained, it is necessary to use a numerical procedure such as the finite difference method to solve a particular partial differential equation. In term of numerical procedure, a particular method can be considered as an efficient method if the method can give an approximate solution within the specified error with the least computational complexity. Throughout this paper, the two-dimensional Porous Medium Equation (2D PME) is discretized by using the implicit finite difference scheme to construct the corresponding approximation equation. Then this approximation equation yields a large-sized and sparse nonlinear system. By using the Newton method to linearize the nonlinear system, this paper deals with the application of the Four-Point Newton-EGSOR (4NEGSOR) iterative method for solving the 2D PMEs. In addition to that, the efficiency of the 4NEGSOR iterative method is studied by solving three examples of the problems. Based on the comparative analysis, the Newton-Gauss-Seidel (NGS) and the Newton-SOR (NSOR) iterative methods are also considered. The numerical findings show that the 4NEGSOR method is superior to the NGS and the NSOR methods in terms of the number of iterations to get the converged solutions, the time of computation and the maximum absolute errors produced by the methods

An unconditionally stable implicit difference scheme for 2D porous medium equations using four-point NEGMSOR iterative method

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 initial boundary 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

Application of MSOR iteration with newton scheme for solutions of 1D nonlinear porous medium equations

This paper considers Newton-MSOR iterative method for solving 1D nonlinear porous medium equation (PME). The basic concept of proposed iterative method is derived from a combination of one step nonlinear iterative method which known as Newton method with Modified Successive Over Relaxation (MSOR) method. The reliability of Newton-MSOR to obtain approximate solution for several PME problems is compared with Newton-Gauss-Seidel (Newton-GS) and Newton-Successive Over Relaxation (Newton-SOR). In this paper, the formulation and implementation of these three iterative methods have also been presented. From four examples of PME problems, numerical results showed that Newton- MSOR method requires lesser number of iterations and computational time as compared with Newton-GS and Newton- SOR methods

Implicit solution of 1d nonlinear porous medium equation using the four-point newton-EGMSOR iterative method

The numerical method can be a good choice in solving nonlinear partial differential equations (PDEs) due to the difficulty in finding the analytical solution. Porous medium equation (PME) is one of the nonlinear PDEs which exists in many realistic problems. This paper proposes a four-point Newton-EGMSOR (4-Newton-EGMSOR) iterative method in solving 1D nonlinear PMEs. The reliability of the 4-Newton-EGMSOR iterative method in computing approximate solutions for several selected PME problems is shown with comparison to 4-Newton-EGSOR, 4-Newton-EG and Newton-Gauss-Seidel methods. Numerical results showed that the proposed method is superior in terms of the number of iterations and computational time compared to the other three tested iterative methods

The application of successive overrelaxation method for the solution of linearized half-sweep finite difference approximation to two-dimensional porous medium equation

Successive overrelaxation or S.O.R. method is a widely known parameter-based iterative method that can regulate a large and sparse system of equations so that the number of iterations required to solve the system can be reduced. Many researchers have applied the S.O.R. method to get the solution to the mathematical problem efficiently. This paper extends the application of the S.O.R. method to solve one of the nonlinear partial differential equation, which is the two-dimensional porous medium equation. The S.O.R. method is incorporated into an iterative method that is formulated based on a half-sweep finite difference approximation, and the Newton-type linearization solves the nonlinear term that presents in the equation. The numerical experiment that uses this innovative numerical method to solve several two-dimensional porous medium equation problems shows significant improvement to the percentage of reduction in the number of iterations and computation time