Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations

We use the general framework of summation by parts operators to construct conservative, entropy-stable and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations and (ii) a recently proposed model by Sv\"ard and Kalisch (2023) with enhanced dispersive behavior. Both models share the property of being conservative in terms of a nonlinear invariant, often interpreted as entropy function. This property is preserved exactly in our novel semidiscretizations. To obtain fully-discrete entropy-stable schemes, we employ the relaxation method. We present improved numerical properties of our schemes in some test cases.