Stiff oscillatory systems, delta jumps and white noise

Abstract

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N >> 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 tends to infinity, and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is O(N). The model problems are integrated numerically in the stiff regime where the time step is of size O(1/N). The convergence of the algorithms is studied in this case in the limit of N tending to infinity and the time step tending to zero. For the white noise problem both strong and weak convergence are considered