An efficient hybrid method based on cubic B-spline and fourth-order compact finite difference for solving nonlinear advection-diffusion-reaction equations

This paper proposes an efficient hybrid numerical method to obtain approximate solutions of nonlinear advection-diffusion-reaction (ADR) equations arising in real-world phenomena. The proposed method is based on finite differences for the approximation of time derivatives while a combination of cubic B-splines and a fourth-order compact finite difference scheme is used for spatial discretization with the help of the Crank-Nicolson method. Since desired accuracy and order of convergence cannot be reached using the traditional cubic B-spline method, to overcome this, the second-order derivatives are approximated using the unknowns and their first derivative approximations with the compact support. Thus, instead of expressing the second-order derivative in second-order accuracy, it is represented by the convergence of order four in the present method. The computed results revealed that this combined approach improves the accuracy of solutions of nonlinear ADR equations in comparison to up-to-date literature even using relatively larger step sizes. Besides, this method is seen to be capable of capturing the behavior of the models with very small viscosity values. The stability of the proposed method has been discussed by considering the matrix stability approach and it has been shown that the method is stable. In addition to the fact that the proposed method obtains sufficiently accurate solutions, another main superiority is its simplicity and applicability, which requires minimum computational effort.