A new compact high order off-step discretization for the system of 2D quasi-linear elliptic partial differential equations

Ranjan K Mohanty,Nikita Setia

doi:10.1186/1687-1847-2013-223

Abstract

A new fourth-order difference method for solving the system of two-dimensional quasi-linear elliptic equations is proposed. The difference scheme referred to as off-step discretization is applicable directly to the singular problems and problems in polar coordinates. Also, new fourth-order methods for obtaining the first-order normal derivatives of the solution are developed. The convergence analysis of the proposed method is discussed in details. The methods are applied to many physical problems to illustrate their accuracy and efficiency. MSC: 65N06.

Highlights

We consider the two-dimensional ( D) quasi-linear elliptic partial differential equation (PDE) of the type a(x, y, u)uxx + b(x, y, u)uyy = f (x, y, u, ux, uy), ( )where (x, y) ∈ R = (, ) × (, ), with boundary ∂R, subject to the Dirichlet boundary conditions given by u(x, y) = v(x, y), (x, y) ∈ ∂R.The PDEs of the type ( ) with variable coefficients model many problems of physical significance
We develop new off-step fourth-order discretizations for the solution of the system of quasi-linear elliptic PDEs with variable coefficients, and the estimates of (∂u/∂n), using the nine grid points of a single computational cell
5 Generalisation of the above methods We extend our methods to the system of D quasi-linear elliptic PDEs of the form: a(i)u(xix) + b(i)u(yiy) = f (i), ≤ i ≤ n for (x, y) ∈ R, with each a(i) = a(i)(x, y, u( ), u( ), . . . , u(n)), b(i) = b(i)(x, y, u( ), u( ), . . . , u(n)) and f (i) = f (i)(x, y, u( ), u( ), . . . , u(n), u(x ), u(x ), . . . , ux(n), u(y ), u(y ), . . . , uy(n)), subject to the Dirichlet boundary conditions given by u(i)(x, y) = v(i)(x, y)

Summary

Introduction

After having determined the fourth-order approximations to the solution of equation ( ), we discuss the fourth-order numerical methods for the estimates of (∂u/∂x) and (∂u/∂y) One may compute these values using the standard central differences: uxl,m = (ul+ ,m – ul– ,m)/( h), uyl,m = (ul,m+ – ul,m– )/( h). 7 Concluding remarks The existing fourth-order nine-point difference methods of [ ] for the numerical solution of the system of second-order quasi-linear D elliptic equations ( ) require a special treatment to handle the numerical scheme at singular points This is because of the appearance of terms, for instance, /(rl– ) for problems in polar coordinates, which would require modification at the singular point l = since r =. Author details 1Department of Applied Mathematics, Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India. 2Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, 110007, India

Methods