A New Fourth-Order Compact Off-Step Discretization for the System of 2D Nonlinear Elliptic Partial Differential Equations

Abstract

This paper discusses a new fourth-order compact off-step discretization for the solution of a system of two-dimensional nonlinear elliptic partial differential equations subject to Dirichlet boundary conditions. New methods to obtain the fourth-order accurate numerical solution of the first order normal derivatives of the solution are also derived. In all cases, we use only nine grid points to compute the solution. The proposed methods are directly applicable to singular problems and problems in polar coordinates, which is a main attraction. The convergence analysis of the derived method is discussed in detail. Several physical problems are solved to demonstrate the usefulness of the proposed methods.

A Fourth Order Difference Method For a Nonlinear Elliptic PDEs in Two Dimension Space

“…To bring out different aspect of the method, we employed it to compute the numerical solution of steady state Navier Stokes equation and convection equation with their constructed exact solutions. For shake of comparison, we have done with methods in [5,6] and find that present method compare favorably. In one considered model problem we find that the computational performance of the method and accuracy depends on the constructed exact solution.…”

“…In tables 4 and 5, we have presented the computed MAU for different values of N and ε. In table, it shows results from [6], for shake of comparison. Problem 4.…”

“…Problem 4. Consider the problem of the nonlinear Navier-Stokes in steady state [6] which,when solving consists of…”

“…To obtain more accurate numerical results several authors proposed improved finite difference method. Recently a compact 9-point fourth order off step finite difference concretization, for solving the system of nonlinear, variable coefficients PDEs and compact 9-point exponential difference method reported in [5,6].…”

A compact high resolution semi-variable mesh exponential finite difference method for non-linear boundary value problems of elliptic nature

“…To resolve this problem, Mohanty and Singh [14] did lot of substantial research. All these schemes [4,8,9] had uniform mesh sizes and needed 5 functional valuations. Using classical second order central difference scheme it is not possible to find the exact solution of most of the equations, in order to get the more appropriate numerical solution, we need to increase the number of mesh points and reduce the size, as a consequence it results in covering extra storage space and thereby escalation in computing time.…”