Two stage waveform relaxation method for the initial value problems of differential-algebraic equations

In this paper, we consider a two stage strategy for waveform relaxation (WR) iterations, applied to initial value problems for differential-algebraic equations (DAEs) in the form Aẏ(t)+By(t)=f(t). Outer iterations of TSWR are defined by MAẏ(k+1)(t)+M1y(k+1)(t)=N1y(k)(t)+NAẏ(k)(t)+f(t), where A=MA−NA, B=M1−N1, and each iteration y(k+1)(t) is computed using an inner iterative process, based on another splitting M1=M2−N2. Meanwhile, by the means of the Theta method, the discretized TSWR of DAEs is realized. Furthermore, when MA is an Hermitian positive semi-definite matrix with P-regular splittings, the convergence and the comparison theorems of TSWR are analyzed. Finally, the numerical experiments are presented.