High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C1C1-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C1C1-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h4,k4)O(h4,k4). Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C1C1-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.