Design of high-order difference scheme and analysis of solution characteristics - Part I: General formulation of high-order difference schemes and analysis of convective stability

In this article, a general design method of second-order difference schemes is presented. By this method, we can easily design any second-order or higher-order difference scheme instead of using complex Lagrange interpolation methods or spline functions. Moreover, it is proved that all existing second-order difference schemes in numerical heat transfer fit this general design style. In addition, based on this general style of second-order scheme, the general style of a second-order absolutely stable scheme is deduced, and the stability definitions are shown in a normalized variable diagram. Finally, through studying the solution characteristics of 14 second-order difference schemes, it is found that, to second order precise, absolutely stable schemes obtained from the general method can achieve good convergence even when the grid Pelect number reaches 100,000. However, at the same time, the false diffusion of the scheme tends to increase along with the increasing value of ai (coefficient in the interface variable definition).