Convergence of parallel multistep hybrid methods for singular perturbation problems

Parallel multistep hybrid methods (PHMs) can be implemented in parallel with two processors, accordingly have almost the same computational speed per integration step as BDF methods of the same order with the same stepsize. But PHMs have better stability properties than BDF methods of the same order for stiff differential equations. In the present paper, we give some results on error analysis of A(α)A(α)-stable PHMs for the initial value problems of ordinary differential equations in singular perturbation form. Our convergence results are similar to those of linear multistep methods (such as BDF methods), i.e. the convergence orders are equal to their classical convergence orders, and no order reduction occurs. Some numerical examples also confirm our results.