A special difference scheme for solving stiff boundary value problems of convective-diffusion transfer

The convective-diffusion transfer equation is often found in problems of hydromechanics and heat and mass transfer. The dominance of convection over diffusion and the change in sign of the coefficient at the first derivative lead to the formation of boundary and internal layers with high gradients of the function. This creates serious difficulties in numerical analysis of the problem using traditional difference schemes. The traditional method of approximating the first derivative using central differences at high Peclet numbers can lead to oscillations and violate the monotonicity of the numerical solution. To avoid this problem, it is necessary to significantly reduce the size of grid cells in narrow areas with large gradients of the unknown function. The use of one-sided differences significantly smears the desired solution, due to the viscosity of the scheme, and leads to loss of accuracy. The practical need to solve stiff boundary value problems requires the development and use of computational technologies that guarantee monotonicity, accuracy, and cost-effectiveness in numerical analysis. In this paper, a new special difference scheme is proposed for the numerical solution of a stiff equation of convective-diffusion transfer. The dominant convective term is eliminated from explicit consideration by transforming the equation into self-adjoined form, which permits the use of well-known numerical approximation techniques. The control volume method is used to construct a difference analogue of a differential equation on a three-point template. The resulting scheme is monotonic and conservative. The test examples show great possibilities of the proposed difference scheme for large Peclet numbers on coarse grids in solving stiff boundary value problems of convective diffusion transfer.