Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre-Green-Naghdi equations

TL;DR

This paper introduces a unified extended Lagrangian framework for multidimensional dispersive wave models, specifically the Serre-Green-Naghdi and Iordanskii-Kogarko-Wijngaarden equations, enabling hyperbolic approximation and effective numerical simulation.

Contribution

It develops a generic dispersive model with a hyperbolic approximation applicable to multiple systems, facilitating numerical analysis of dispersive waves.

Findings

The new model is unconditionally hyperbolic for SGN and IKW systems.

Numerical simulations show close agreement with exact solutions.

The approach effectively captures dispersive shock waves.

Abstract

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