Dimitrios Mitsotakis, David I. Ketcheson, and I have published our new paper A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations on arXiv.
We develop general tools to construct fully-discrete, conservative numerical methods and apply them to several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations are built using the unifying framework of summation by parts operators and include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. Classical time integration schemes such as Runge-Kutta methods and linear multistep methods are applied and the recent relaxation technique is used to enforce temporal conservation.