Half boundary method for steady state convection–diffusion equations with different boundary conditions

Convection-diffusion equation is widely used in many fields of science, technology and engineering. The calculation is usually difficult and time-consuming. In this paper, a highly efficient meshless boundary type method called the half boundary method (HBM) is applied to solve convection diffusion equation. The main idea of HBM is to reduce the order of the convection–diffusion equations by introducing new variables and construct the relations of the variables between the nodes inside the area and the nodes on half of the boundaries by using integration, inverse matrix and reasonable transformation. Using the relations, the temperature and heat flux at any point can be calculated simultaneously after obtaining the variables on half of the boundaries according to the boundary conditions. Given that unknown variables exist only on half of the boundaries, the internal storage in the HBM is less than that required in the finite volume method when the division number is large. And it doesn't need shape functions in the finite element method or a fundamental solution in the boundary element method. HBM can obtain very precise results when solving the convection dominated problems. The validity and accuracy of the proposed method are authenticated by the numerical examples.