Geometric Numerical Integration

This chapter deals with numerical methods for second-order differential equations with oscillatory solutions. These methods are designed to require a new complete function evaluation only after a time step over one or many periods of the fastest oscillations in the system. Various such methods have been proposed in the literature some of them decades ago, some very recently, motivated by problems from molecular dynamics, astrophysics and nonlinear wave equations. For these methods it is not obvious what implications geometric properties like symplecticity or reversibility have on the long-time behaviour, e.g., on energy conservation. The backward error analysis of Chap. IX, which was the backbone of the results of the three preceding chapters, is no longer applicable when the product of the step size with the highest frequency is not small, which is the situation of interest here. The “exponentially small” remainder terms are now only O(1)! At least for a class of nonlinear model problems, which includes the Fermi-Pasta-Ulam problem of Sect. 1.4.1, a substitute for the backward error analysis of Chap. IX is given by the modulated Fourier expansions of the exact and the numerical solutions. Among other properties, they permit us to understand the numerical long-time conservation of energy (or the failure of conserving energy in certain cases). It turns out, symmetry of the methods is still essential, but symplecticity plays no role in the analysis and in the numerical experiments, and new conditions of an apparently non-geometric nature come into play.