Stiff Oscillatory Systems, Delta Jumps and White Noise

Abstract

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of $N \gg 1$ 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 $N \to \infty$ and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is <e6>OM</e6>(N). The model problems are integrated numerically in the stiff regime where the time-step $\Delta t$ satisfies $N \Delta t={\cal O}(1).$ The convergence of the algorithms is studied in this case in the limit N → ∞ and Δt → 0.For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically.