Stiff Oscillatory Systems, Delta Jumps and White Noise

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N À 1 harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as N → ∞ and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators isO(N ).The model problems are integrated numerically in the stiff regime where the time-step1t satisfies N1t = O(1). The convergence of the algorithms is studied in this case in the limit N →∞ and 1t → 0. For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically. Date received: August 25, 1999. Final version received: May 3, 2000. Communicated by Arieh Iserles. AMS classification: 34A65, 60H10, 65L70, 82C80. 70 B. Cano, A. M. Stuart, E. Süli, and J. O. Warren