Numerical behaviour of stable and unstable solitary waves
The authors discuss the numerical solution of the generalised Benjamin-Bona-Mahony (BBM) equation \(u_t+u_x+(u^p)_x-u_{txx}=0\) with an integer \(p>1\). In contrast to the classical BBM equation (p=2), it is in general not a completely integrable system. However, it is still a Hamiltonian system invariant under a one-parameter group of translations and admits a two-parameter family of solitary wave solutions that may be interpreted as relative equilibria associated with this symmetry group. For \(p>5\) the (orbital) stability of these solitary waves depends on the velocity: in a certain range the waves are linearly unstable. The main goal of the article is to show that numerical methods that preserve some conserved quantities are more appropriate for this problem. The authors compare the use of two different Runge-Kutta methods for the time integration: the implicit midpoint rule preserving the first integral \(I\) and a third order diagonally implicit method which preserves neither the Hamiltonian \(H\) nor \(I\). For the spatial discretisation a Fourier pseudospectral method is applied so that essentially all numerical errors are due to the time integration. The results show clearly that the conservative method is superior despite its lower order. The phase shift is much smaller and the error in the amplitude is bounded; the integration error grows only linearly in contrast to a quadratic growth for the third order method. The main part of the article is devoted to a detailed error analysis in order to explain this behaviour. The analysis is based on the construction of an asymptotic expansion in which a so-called modified solitary wave appears which is perturbed in the basic parameters: amplitude, velocity and phase. It turns out that the classical approach already applied to a number of other equations does not well explain the experimental observations. The authors derive a new modified solitary wave that better explains the observations.
generalised Benjamin-Bona-Mahony equation
scientific article; zbMATH DE number 1787725
scientific article