Stability of Backward Euler Multirate Methods and Convergence of Waveform Relaxation

For a large class of traditional backward Euler multirate methods we show that stability is preserved when the methods are applied to certain stable (but not necessarily monotonic) non-linear systems. Methods which utilize waveform relaxation sweeps are shown to be stable and converge for certain monotonic systems. 1 Relaxing the monotonicity condition Consider the system of ODE’s (i.e. ordinary differential equations) y = f(t, y) for t ≥ t0, where y(t) ∈ <. (1.1) In order to prove stability of numerical methods applied to (1.1), some type of monotonicity condition is usually imposed on the system of ODE’s, i.e. a condition ensuring the existence of a norm such that ‖ u(t2) − v(t2) ‖≤‖ u(t1) − v(t1) ‖ for t2 ≥ t1 ≥ t0, (1.2) holds for any two solutions u and v of (1.1). Most often it is assumed that the norm can be chosen as an inner-product norm, but an example by Spijker ([1, p. 658]): y = ( −1 (1+ | t |)−1 0 0 ) y, (1.3) shows that some monotonic systems are excluded by that assumption. On the other hand, it is easy to see that for certain monotonic systems the norm has to be an inner-product norm: y = Q ( 0 −α α 0 ) Q y, where Q is an orthogonal matrix. (1.4) Published in BIT 32 (1992), 350-366. Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK–2100 Copenhagen, Denmark 1 Since the solutions of (1.4) are periodic, ‖ u(t) − v(t) ‖ (cf. (1.2)) must be constant and thus the unit sphere of the norm has to be proportional to the orbits (being ellipses). Although (1.3) is not monotonic in any inner-product norm, it is, however, monotonic in a differentiable norm: ‖ (