Layer-adapted Numerical Methods For Parabolic Singularly Perturbed Convection-diffusion Problems

Singular perturbation problems are widely applied in science,technology and engineering fields,including porous media seepage,river network water quality,Black-Scholes model in financial,etc.As is known,classical finite difference schemes cannot solve the wild oscillations occuring nearby the boundary layer,but layer-adapted graded mesh method can do well.So how to solve the singular perturbation problems numerically by using layer-adapted meshes has become a popular research topic.This paper studies the one-dimensional parabolic singularly perturbed convection-diffusion problems and we mainly do the following work:In the first part,we give the properties of the solution and construct a fully discrete scheme using the Crank-Nicolson method on the uniform mesh for the time discretization and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh for the spatial discretization.Then we derive the important discrete comparison principle.The proposed scheme is proved to be uniform convergent having order O(M^-2)in time and order O(N^-2)on the regular part and order O?N-1?on the layer part in space by means of truncation error,barrier functions and so on.Finally,the singular perturbation equations where the exact solutions are known are selected for numerical experiments,and the numerical results demonstrate the error estimate.In the second part,we derive the further properties of the solution and construct a fully discrete scheme using the Crank-Nicolson method on the uniform mesh to discretize in time and the hybrid difference that combines the midpoint upwind difference scheme on the regular part and the central difference scheme on the layer part on the Shishkin mesh in space.Similarly,we obtain the important discrete comparison principle.Further,the fully discrete scheme is proved to be uniform with respect to the small diffusion parameter having order O(M^-2)in time and order O(N^-2)on the regular part and O(N^-2ln² N)on the layer part in space.Finally,the singular perturbation equations where the exact solutions are known or unknown are selected for numerical experiment,and the numerical results demonstrate the effectiveness of the proposed scheme and the correctness of its error estimate.