Using the incomplete ILU decomposition for convection-diffusion processes modeling in anisotropic media

Three-dimensional stationary convection-diffusion problem with mixed derivatives, describing convective-diffusion processes in anisotropic media, is considered. The problem solution in an anisotropic medium is of considerable interest for various models. The upwind finite-difference approximation of this problem is held at thirteen point pattern. Sufficient conditions for the matrix of the system of linear algebraic equations (SLAE), obtained after the natural ordering of nodes of the grid area, to be an M-matrix, are proved. The method of ILU decomposition, which converges for M-matrices, is used for the numerical solution of SLAE. Numerical experiments, which showed the effectiveness of the method for solving convection-diffusion problem in both isotropic and anisotropic medium, have been carried out. The disadvantage of the used method is the restriction on the coefficients on mixed derivatives. Obtained as a result of theoretical studies, this restriction is a part of a set of sufficient conditions for M-matrix operator.