Numerical solution of highly oscillatory ordinary differential equations

Abstract

One of the most difficult problems in the numerical solution of ordinary differential equations (ODEs) and in differential-algebraic equations (DAEs) is the development of methods for dealing with highly oscillatory systems. These types of systems arise, for example, in vehicle simulation when modelling the suspension system or tyres, in models for contact and impact, in flexible body simulation from vibrations in the structural model, in molecular dynamics, in orbital mechanics, and in circuit simulation. Standard numerical methods can require a huge number of time-steps to track the oscillations, and even with small stepsizes they can alter the dynamics, unless the method is chosen very carefully.

Multi-revolution composition methods for highly oscillatory differential equations

“…Consider y = (q 1 , q 2 , q ′ 1 , q ′ 2 ) the exact solution of (24) where we assume that the vector fields g 1 , g 2 have the regularity C p+1 . Consider ϕ ε the flow map with time ε of (10)- (25) where η = ε is fixed, and Ψ N,H a MRCM (7) satisfying the hypotheses of Theorem 2.6. Then, there exist C, H 0 > 0 such that for all H ≤ H 0 , N ≥ 2s, and mH ≤ T ,…”

“…A class of multi-revolution Runge-Kutta type methods has then been studied in the context of oscillatory problems of the form (10) in [3,4,23,25] and [2] was the first systematic analysis of the order conditions of convergence. Closely related methods were considered in [19] and also in [5].…”

A Gautschi-type method for oscillatory second-order differential equations

“…For a recent survey article on existing numerical approaches to oscillatory differential equations we refer to [9].…”

A Stroboscopic Numerical Method for Highly Oscillatory Problems

“…The approach suggested here is related to methods called envelop-following or multi-revolution (see [12,2] and their references) that go back to the 1960's and have been successfully used in a number of application areas, including celestial mechanics and circuit theory. Note that, while in this paper both the macro-and micro-integrators are standard ODE solvers, the multi-revolution technique requires the construction of new special formulae.…”