Tailored Finite Point Method for Parabolic Problems

In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.