Rethinking to Finite Difference Time-Step Integrations

The numerical time step integrations of PDEs are mainly carried out by the finite difference method to date. However,when the time step becomes longer, it causes the problem of numerical instability,. The explicit integration schemes derived by the singlepoint precise integration method given in this paper are proved unconditionally stable.Comparisons between the schemes derived by the finite difference method and the schemes by the method employed in the present paper are made for diffusion and convective-diffusion equations. Nunierical examples show the superiority of the singlepoint integration method.