An unconditionally stable parallel difference scheme for parabolic equations

Abstract

An unconditionally stable difference scheme with intrinsic parallelism is constructed in this paper, taking the parabolic equation u t = u xx as an example. The main ideas are described as follows. We decompose the domain Ω into some overlapping subdomains, take values of the last time layer as values of the time layer on inner boundary points of subdomains, solve it with the fully implicit scheme inside each subdomain, and then take correspondent values of its neighbor subdomains as their values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points. Its unconditional stability and convergence are proved. Though its truncation error is O(1), we will still prove that the estimate order is improved to O( τ+ h).