Numerical stroboscopic averaging for ODEs and DAEs

Maripaz Calvo (1) , Philippe Chartier (2) , Ander Murua (3) , Jesus Maria Sanz-Serna (4)

1 MA - Departamento de Matemática Aplicada
2 IPSO - Invariant Preserving SOlvers
3 Konputazio Zientziak eta A.A. Saila [Spain]
4 MA - Departamento de Matemática Aplicada [Valladolid]

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt = f(y; t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt = F(Y ) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LISP algorithms of Kirchgraber and multirevolution methods.

Mots clés: Highly oscillatory problems; Stroboscopic averaging; Multirevolution methods; Differential algebraic equations

Domaines: Analyse numérique [math.NA]

Maripaz Calvo, Philippe Chartier, Ander Murua, Jesus Maria Sanz-Serna. Numerical stroboscopic averaging for ODEs and DAEs. Applied Numerical Mathematics, 2011, 61 (10), pp.1077-1095. ⟨10.1016/j.apnum.2011.06.007⟩. ⟨hal-00777181⟩