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.

翻译：我们利用求和分部算子的一般框架，为两个具有变化地形的非线性色散浅水波方程组构建保守、熵稳定且平衡的半离散格式：（i）耦合Benjamin-Bona-Mahony（BBM）方程的一个变体，以及（ii）Sv\"ard和Kalisch（2023）最近提出的具有增强色散行为的模型。这两个模型均具有非线性不变量（常被解释为熵函数）的保守性质，该性质在我们全新的半离散格式中被精确保持。为获得全离散熵稳定格式，我们采用松弛方法。我们在若干测试案例中展示了所提格式改进的数值性质。