A fast method for solving time-dependent nonlinear convection diffusion problems

In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.