On the Convergence of Non-Polynomial Spline Finite Difference Method for Quasi-Linear Elliptic Boundary Value Problems in Two-Space Dimensions

An approximate solution technique based on non-polynomial spline and finite difference method has been described for two-space dimensional quasi-linear elliptic boundary value problems. The numerical scheme in the limiting case of non-polynomial spline parameter provides the cubic spline scheme. The proposed scheme is analyzed for the convergence using matrix theory. Experimental results show the importance of using non-polynomial spline scheme over corresponding cubic spline scheme in terms of iterations to achieve desired accuracy. The method is tested for the convergence and corroborates the theoretical truncation errors. The accuracy in the solutions is obtained for Tricomi and Khokhlov-Zabolotskaya equations for various mesh step sizes.