Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation

In this paper we investigate the parallel difference schemes of parabolic equation, in particular, two kinds of difference schemes with intrinsic parallelism are constructed. Firstly we combine the values of previous two time levels at the interface points to get the (Dirichlet) boundary condition for the sub-domain problems. Then the values in the sub-domains are calculated by fully implicit scheme. And then finally the values at the interface points are computed by fully implicit scheme. The unconditional stability of these schemes is proved, and the convergence rate of second order is also obtained. Numerical results are presented to examine the accuracy, stability and parallelism of the parallel schemes.