A conservative fully-discrete numerical method for the regularised
shallow water wave equations

The paper proposes a new, conservative fully-discrete scheme for the
numerical solution of the regularised shallow water Boussinesq system of
equations in the cases of periodic and reflective boundary conditions. The
particular system is one of a class of equations derived recently and can be
used in practical simulations to describe the propagation of weakly nonlinear
and weakly dispersive long water waves, such as tsunamis. Studies of
small-amplitude long waves usually require long-time simulations in order to
investigate scenarios such as the overtaking collision of two solitary waves or
the propagation of transoceanic tsunamis. For long-time simulations of
non-dissipative waves such as solitary waves, the preservation of the total
energy by the numerical method can be crucial in the quality of the
approximation. The new conservative fully-discrete method consists of a
Galerkin finite element method for spatial semidiscretisation and an explicit
relaxation Runge--Kutta scheme for integration in time. The Galerkin method is
expressed and implemented in the framework of mixed finite element methods. The
paper provides an extended experimental study of the accuracy and convergence
properties of the new numerical method. The experiments reveal a new
convergence pattern compared to the standard, non-conservative Galerkin
methods.