Nonconservative discontinuous Galerkin methods for shallow water moment models

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2021-12
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Logemann, Caleb
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Rossmanith, James
Liu, Hailiang
Luo, Songting
Rothmayer, Alric
Yan, Jue
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Mathematics
Abstract
The shallow water equations are a thin-film approximation of the more complex three-dimensional incompressible Navier-Stokes system that are valid for wavelength phenomena that are large relative to the characteristic vertical length scale. Such equations are ubiquitous in simplified models of the atmosphere, oceans, lakes, and rivers; however, their main shortcoming is that all vertical variation in the solution is lost. In order to overcome this shortcoming, several models have been introduced that allow for some limited resolution of the vertical structure. A notable set of such models are the so-called {\it shallow water moment equations} first introduced by Kowalski and Torrilhon [{\it Moment Approximations and Model Cascades for Shallow Flow}, Communications in Computational Physics (2019)]. These shallow water moment equations introduce vertical moments into the shallow flow's horizontal velocity profile, which allows for more complicated and physically relevant fluid flows to be represented. Similar to other generalizations of the shallow water equations such as the multi-layer shallow water models, the shallow water moment equations for two or more moments are not globally hyperbolic. In order to overcome this hyperbolic deficiency, Koellermeier and Pimentel-Garc\'ia [{\it Steady states and well-balanced schemes for shallow water moment equations with topography.}, \url{https://arxiv.org/abs/2011.07667} (2020)] introduced a variation of the shallow water moment model, which is referred to as the {\it shallow water linearized moment equations} and showed that this model is globally hyperbolic in one dimension. In this work we extend these models to higher-dimension, and show that even in this case these models remain hyperbolic. A further challenge with these models is that the resulting system of partial differential equations are {\it nonconservative hyperbolic systems}; and thus, weak solutions cannot be described using the traditional theory of distributions. In this work, we use the Dal Maso, Le Floch, and Murat [{\it Definition and weak stability of nonconservative products}, Journal de Math{\'e}matiques Pures et Appliqu{\'e}es, (1995)] theory of nonlinear hyperbolic systems in nonconservative form to describe solutions of these equations. We then use this theoretical framework to develop novel high-order discontinuous Galerkin methods for both the shallow water moment models and the linearized moment model variants. The resulting discontinuous Galerkin scheme is implemented for problems in one and two dimensions using both Cartesian and triangular unstructured meshes. The resulting numerical schemes are high-order accurate up to fifth order in both space and time. Finally, with an eye towards large scale atmospheric modeling, we develop in this work a variant of the shallow water moment equations on the surface of the sphere.
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