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 .
We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for 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 include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods . We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests .
R
general tools to construct fully-discrete,
a general framework for designing
R
and apply them to
based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for
D
are built using the unifying framework of summation by parts operators and
R
Classical time integration schemes such as
The time discretization is essentially explicit, using relaxation
R
and linear multistep methods are applied and the recent relaxation technique is used to enforce temporal conservation
. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests