A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations (Communications in Computational Physics, 2021)
We develop a general framework for designing conservative numericalmethods based on summation by parts operators and split forms in space, combinedwith relaxation Runge-Kutta methods in time. We apply this framework to createnew classes of fully-discrete conservative methods for several nonlinear dispersivewave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm,Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizationsconserve all linear invariants and one nonlinear invariant for each system. The spatialsemidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentiallyexplicit, using relaxation Runge-Kutta methods. We implement some specific schemesfrom among the derived classes, and demonstrate their favorable properties throughnumerical tests.