论文标题
用于多维色散波的数值研究的扩展拉格朗日方法：应用于Serre-Green-Naghdi方程的应用
Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre-Green-Naghdi equations
论文作者
论文摘要
在本文中，我们研究了两个多维非线性分散系统：描述分散浅水流的serre-green-naghdi（SGN）方程，以及描述包含小型压缩气泡的流体的Iordanskii-Kogarko-Wijngaarden（IKW）方程。这些模型是给定Lagrangian的Euler-Lagrange方程，共享共同的数学结构，即压力对宏观变量材料衍生物的依赖性。我们开发了一个通用的分散模型，这样，如果仅指定适当的拉格朗日，那么SGN和IKW系统将成为其特殊情况，然后使用Favrie和Gavrilyuk（2017）中提出的扩展Lagragian方法来构建其双曲线近似。对于SGN和IKW病例，新的近似模型都是无条件的双曲线，并准确地描述了分散现象，该现象允许施加不连续的初始数据和研究分散性冲击波。我们将SGN系统的二维双曲版本视为数值模拟的示例，并应用二阶隐式解释方案以数值整合系统。获得的1-D和2-D结果与可用的精确解决方案和数值测试密切一致。 \ textbf {关键字：}分散浅水方程，起泡的流体，欧拉 - 拉格朗日方程，双曲线保护法，多维波，隐式 - 明确的数值方法
In this paper we study two multidimensional nonlinear dispersive systems: the Serre-Green-Naghdi (SGN) equations describing dispersive shallow water flows, and Iordanskii-Kogarko-Wijngaarden (IKW) equations describing fluids containing small compressible gas bubbles. These models are Euler-Lagrange equations for a given Lagrangian and share common mathematical structure, namely the dependence of the pressure on material derivatives of macroscopic variables. We develop a generic dispersive model such that SGN and IKW systems become its special cases if only one specifies the appropriate Lagrangian, and then use the extended Lagragian approach proposed in Favrie and Gavrilyuk (2017) to build its hyperbolic approximation. The new approximate model is unconditionally hyperbolic for both SGN and IKW cases, and accurately describes dispersive phenomena, which allows to impose discontinuous initial data and study dispersive shock waves. We consider the 2-D hyperbolic version of SGN system as an example for numerical simulations and apply a second order implicit-explicit scheme in order to numerically integrate the system. The obtained 1-D and 2-D results are in close agreement with available exact solutions and numerical tests. \textbf{Keywords: } dispersive shallow water equations, bubbly fluids, Euler-Lagrange equations, hyperbolic conservation laws, multidimensional waves, implicit-explicit numerical methods
