Convergence conditions on waveform relaxation of general differential-algebraic equations

In the paper, we show some new convergence conditions on waveform relaxation (WR) for general differential-algebraic equations (DAEs). The main conclusion is that the convergence conditions on index r+1 can be derived from that of index r, in which the corresponding system is composed by ordinary differential equations if r=0. The approach of analysing relaxation process is novel for WR solutions of DAEs. It is also the first time to give the convergence conclusions for general index systems of DAEs in the WR field.