The mass-preserving and modified-upwind splitting DDM scheme for time-dependent convection–diffusion equations

Abstract

In the paper, a new mass-preserving and modified-upwind S-DDM scheme over non-overlapping multi-block sub-domains for solving time-dependent convection–diffusion equations in two dimensions is developed and analyzed. On each sub-domain, the intermediate fluxes on the interfaces of sub-domains are firstly computed by the modified semi-implicit flux schemes. Then, the solutions and fluxes in the interiors of sub-domains are computed by the modified-upwind splitting implicit solution and flux coupled schemes. On the theoretical aspect, analyzing the mass-preserving and modified-upwind S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique, the conservative modified-upwind technique, and the domain decomposition. We prove the proposed scheme to be mass conservative and unconditionally stable in discrete L2-norm and further prove its convergence and obtain error estimates. Numerical experiments are presented to illustrate mass conservation, convergence and parallelism.