A half boundary method for two dimensional unsteady convection–diffusion equations

Abstract

Convection diffusion equation is widely applied in many fields of science and technology. Many practical engineering problems can be expressed by this equation in unsteady state. However, it is usually difficult and time-consuming to find its solution. In this paper, a boundary type method named half boundary method (HBM) is proposed for two dimensional unsteady convection–diffusion equations. The main idea of HBM is introducing new variables to reduce the order of the equation and build relations of variables between nodes inside the area and nodes at half of the boundaries, namely the unknown variables. Since only variables at half of the boundaries are chosen as unknown variables, using HBM can realize dimensionality reduction and the maximum matrix order is less than that in finite volume method when considering large division number, which makes HBM fast and efficient. After the unknown variables are obtained by the boundary conditions, all variables at nodes can be obtained simultaneously according to the relations mentioned above. Numerical studies are carried out for convection dominated problems, problems with variable or discontinuous coefficient and problems with mixed boundary conditions, which show the validity of HBM for two-dimensional unsteady convection-diffusion problems and high accuracy for convection-domination problems.