A block-centered characteristic finite difference method for convection-dominated diffusion equation

In this paper, a block-centered characteristic finite difference method is proposed for solving the convection-dominated diffusion equation on nonuniform grids. The resulting scheme is first-order accurate in time and second-order accurate in space. The stability analysis and error estimate are deduced rigorously. And the numerical solutions of unknown variable along with its first derivatives are also obtained. Finally, numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the proposed method. Moreover, we show that numerical results using non-uniform grids are significantly more accurate than those using uniform grids for high gradient problems.