A Multigrid Method of the Second Kind for Solving Linear Systems of Odes Discretized by Continuous Runge-Kutta Methods

Abstract

A Waveform Relaxation method as applied to a linear system of ODEs is the Picard iteration for a linear Volterra integral equation of the second kind ({\\cal I} - {\\cal K})y = b \\eqno (1) called Waveform Relaxation second kind equation. A corresponding Waveform Relaxation Runge-Kutta method is the Picard iteration for a discretized version ({\\cal I} - {\\cal K}_l )y_l = b_l \\eqno (2) of the integral equation (1), where y l is the continuous solution of the original linear system of ODE provided by the so called limit method. We consider a W-cycle multigrid method, with Picard iteration as smoothing step, for iteratively computing y l . This multigrid method belongs to the class of multigrid methods of the second kind as described in Hackbusch [3, chapter 16]. In the paper we prove that the truncation error after one iteration is of the same order of the discretization error y l @ y of the limit method and the truncation error after two iterations has order larger than the discretization error. Thus we can see the multigrid method as a new numerical method for solving the original linear system of ODE which provides, after one iteration, a continuous solution of the same order of the solution of the limit method, and after two iterations, a solution with asymptotically the same error of the solution of the limit method. On the other hand the computational cost of the multigrid method is considerably smaller than the limit method.