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 → ∞ 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 Δt satisfies NΔt=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.