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 A y ̇ ( t ) + B y ( t ) = f ( t ) . Outer iterations of TSWR are defined by M A y ̇ ( k + 1 ) ( t ) + M 1 y ( k + 1 ) ( t ) = N 1 y ( k ) ( t ) + N A y ̇ ( k ) ( t ) + f ( t ) , where A = M A − N A , B = M 1 − N 1 , and each iteration y ( k + 1 ) ( t ) is computed using an inner iterative process, based on another splitting M 1 = M 2 − N 2 . Meanwhile, by the means of the Theta method, the discretized TSWR of DAEs is realized. Furthermore, when M A 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.