Nonlinear Galerkin Method for Hamiltonian Systems

Abstract

In its simplest form, the Galerkin method is the truncation of a differential equation by projection on a set of (spatial) base functions, in the present case: truncation to a certain number n of Fourier modes. But rather than neglecting all other modes completely, the nonlinear modification consists in taking some of the effects of the higher modes into account in the calculation of the first n modes. Specifically, if in the dynamic equations for the higher modes the time-derivative is set equal to zero, the equations simplify to quasi-stationary relations from which the higher modes can be solved, as function of the lower modes.

In dissipative systems this procedure is motivated by the very fast decay of higher modes. In the present paper, however, we apply the idea on Hamiltonian, i.e. conservative, systems. With the Korteweg-de Vries equation as an example, we will consider the direct truncation and a Nonlinear Galerkin method, both in Hamiltonian formulation. The accuracy of both methods is analysed in terms of negative powers of n, and comparisons are visualised graphically.