Waveform Relaxation Methods for Functional Differential Systems of Neutral Type

Abstract

We investigate continuous-time and discretized waveform relaxation iterations for functional differential systems of neutral type. It is demonstrated that continuous-time iterations converge linearly for neutral equations and superlinearly when the right hand side is independent of the history of the derivative of the solution. The error bounds for discretized iterations are also obtained and some implementation aspects are discussed. Numerical results are presented which indicate a potential speedup of this technique as compared with the classical approach based on discrete variable methods.

Remarks on evaluation of spectral radius of operators arising in WR methods for linear systems of ODEs

“…It is obvious that the unique solution of problem (1) exists. The same is true for problem (4), which guarantees the existence of the sequence (x k ). Our problem is to find conditions for the convergence of the WR sequence.…”

“…Another proof follows from rather non-trivial Theorem 2 in [4] and the standard theorem of the analysis on the relation between Cauchy and d'Alambert criteria for series convergence (see, for example [21]; also notice that the assertion of Theorem 2 in [4] holds for any K > 0).…”

“…However, the method was born much earlier, see, for instance, the papers [10][11][12] as well as a very general formulation due to Ważewski [27,28] and Kwapisz [13,14]. A number of papers discuss in details the convergence and error estimates of various WR methods (see, for example, [1][2][3][4]7,[17][18][19][20]25,29] and many others).…”

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

“…We also refer to [2,3] for the discussion of various acceleration methods of waveform relaxation iterations. These techniques in the context of functional-differential equations are discussed in [12,29,28]. Waveform relaxation methods for differential-algebraic systems are examined in [11].…”

Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations

“…[9] in which Enright and Hu considered the convergence of continuous Runge-Kutta methods for NVDIDEs y (t) = f (t, y(t)) + t t−τ K(t, θ, y(θ), y (θ))dθ, t 0. (1.5) In particular, Jackiewicz [13][14][15] systematically investigated the convergence of various numerical methods for more general neutral functional differential equations (NFDEs). However, these important convergence results are based on the classical Lipschitz conditions.…”