A conservative parallel difference method for 2-dimension diffusion equation

In this paper, a conservative parallel difference scheme, which is based on domain decomposition method, for 2-dimension diffusion equation is proposed. In the construction of this scheme, we use the numerical solution on the previous time step to give a weighted approximation of the numerical flux. Then the sub-problems with Neumann boundary are computed by fully implicit scheme. What is more, only local message communication is needed in the program. We use the method of discrete functional analysis to give the proof of the unconditional stability and second-order convergence accuracy. Some numerical tests are given to verify the theory results.