A Compact Finite Difference Scheme for Reaction-convection-diffusion Equation

In this paper, a new method is developed for approximating solution to the reaction-convection-diffusion equation, in which reaction rate and diffusion coefficient are small parameters. A compact finite difference scheme (CFD) is applied for discretizing spatial derivatives of linear reaction-convection-diffusion equation, which leads to a linear system of ordinary differential equations. To solve the resulted system, the cubic C-1-spline collocation method is applied. The accuracy in space and time is of fourth-order i.e. O(h(4), k(4)). Although the proposed scheme is not A-stable, it is shown to be unconditionally stable. Numerical results show that the combination of the compact finite difference approximation and the cubic C-1-spline collocation methods give an efficient method for solving the reactionconvection-diffusion equation.