A HBM approach for temperature and heat flux convection-diffusion equations and nonlinear problems

Solving convection diffusion equation is widely required in many fields of science, technology and engineering. The calculation is usually difficult and time-consuming. In this paper, a highly efficient method, the half method (HBM), is proposed to solve the convection diffusion equation. The main idea of HBM is to reduce the order of the convection-diffusion equations by introducing a new variable and constructing the relations of the variables between the nodes inside the area and the nodes on half of the boundaries. Using the relations, the temperature and heat flux at any point can be calculated simultaneously, after obtaining the variables on the half of the boundaries. Because the unknown variables exist on only half of the boundaries, the computing matrix is reduced to only second order regardless of the number of nodes, the internal storage in the HBM is even less than that required in the finite volume method, making the HBM extremely fast and efficient. The validity and accuracy of the proposed method are investigated. Numerical studies for steady and unsteady, as well as nonlinear convection-diffusion equations, were carried out. The results show that HBM is more accurate than the finite volume method under identical grids conditions.