Cubic Spline Approximation for Two-Dimensional Nonlinear Elliptic Boundary Value Problems

Abstract

We report a new 9-point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two-dimensional nonlinear elliptic partial differential equations of the form

$$ A\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial x^{2} }} + B\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial y^{2} }} = f\left( {x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}} \right),\left( {x,y} \right) \in\Omega $$

defined in the domain $\Omega = \left\{ {\left( {x,y} \right):0 < x,y < 1} \right\}$ with boundary $\partial\Omega$, where $A\left( {x,y} \right) > 0$ and $B\left( {x,y} \right) > 0$ in $\Omega$. The corresponding Dirichlet boundary conditions are prescribed by

$$ u\left( {x,y} \right) = \psi \left( {x,y} \right),\quad \left( {x,y} \right) \in \partial\Omega $$

The main spline relations are presented and incorporated into solution procedures for elliptic partial differential equations. Available numerical methods based on cubic spline approximations for the numerical solution of nonlinear elliptic equations are of second-order accurate. Although 9-point finite difference approximations of order four accurate for the solution of nonlinear elliptic differential equations are discussed in the past, but these methods require five evaluations of the function f. In this piece of work, using the same number of grid points and three evaluations of the function f, we have derived a new stable cubic spline method of order 2 in y- and order 4 in x-directions for the solution of nonlinear elliptic equation. However, for a fixed parameter (Δy/Δx ²), the proposed method behaves like a fourth order method. The accuracy of the proposed method is exhibited from the computed results. The proposed method is applicable to Poisson’s equation and two-dimensional Navier-Stokes’ equations of motion in polar coordinates, which is main highlight of the work. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.