NUMERICAL SOLUTION OF CONVECTION-DIFFUSION EQUATION USING CUBIC B-SPLINES COLLOCATION METHODS WITH NEUMANNS BOUNDARY CONDITIONS

Abstract

In this paper, two numerical methods are proposed to approximate the solutions of the convection-diffusion partial differential equations with Neumann boundary conditions. The methods are based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. In Method-I, we discretize the time derivative with Crank Nicolson scheme and handle spatial derivatives with cubic B-splines. Stability of this method has been discussed and shown that it is unconditionally stable. In Method-II, we apply cubic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK54 scheme. These methods needs less storage space that causes to less accumulation of numerical errors. In numerical test problems, the performance of these methods is shown by computing for different time levels. Illustrative five examples are included to demonstrate the validity and applicability of these methods. Results shown by these methods are found to be in good agreement with the exact solutions. Easy and economical implementation is the strength of these methods.