A stochastic Galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve hyperbolicity of the original system. In this work, we develop hyperbolicity-preserving stochastic Galerkin formulation for the one-dimensional shallow water equations by carefully selecting the polynomial chaos expansion of the nonlinear $q^2/h$ term in terms of the polynomial chaos expansions of the conserved variables. In addition, in an arbitrary finite stochastic dimension, we establish a sufficient condition to guarantee hyperbolicity of the stochastic Galerkin system through a finite number of conditions at stochastic quadrature points. Further, we develop a well-balanced central-upwind scheme for the stochastic shallow water model and derive the associated hyperbolicty-preserving CFL-type condition. The performance of the developed method is illustrated on a number of challenging numerical tests.
Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadratu…
Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a modification of this method that uses a slope limiter to retain admissible solutions of the system, while providing high-order approximations in the physical and stochastic space. This is done using a spatial discontinuous Galerkin scheme and a Multi-Element stochastic Galerkin ansatz in the random space. We analyze the convergence of the resulting scheme and a…
On Hyperbolic Stochastic Galerkin Projections of Shallow Water Linearised Moment Equations
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；CogniGron (Groningen Cognitive Systems and Materials Centre), University of Groningen（格罗宁根大学认知系统材料中心（CogniGron））；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学与统计系）
High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuo…
A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density
Chongqing University（重庆大学）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；School of arts and sciences, Chengdu College of University of Electronic Science and Technology of China（电子科技大学成都学院文理学院）；Sichuan International Studies University（四川外国语大学）
In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to machine error. Another feature is the non-oscillatory property, which is very important in the numerical simulation when there exist some shock discontinuities. To control the spurious oscillations, we construct an OFDG method with an extra damping term to the existing well-balanced …
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
Universität zu Köln（科隆大学）；The Florida State University（佛罗里达州立大学）
We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and compare the presented scheme to an entropy stable scheme based on simplicial summation-by-parts (SBP) finite difference operators. Finally, we report the computational performance of an implementa…
Stochastic Galerkin methods can quantify uncertainty at a fraction of the computational expense of conventional Monte Carlo techniques, but such methods have rarely been studied for modelling shallow water flows. Existing stochastic shallow flow models are not well-balanced and their assessment has been limited to stochastic flows with smooth probability distributions. This paper addresses these limitations by formulating a one-dimensional stochastic Galerkin shallow flow model using a low-order Wiener-Hermite Polynomial Chaos expansion with a finite volume Godunov-type approach, incorporating…
In this article we study intrusive uncertainty quantification schemes for systems of conservation laws with uncertainty. Standard intrusive methods lead to oscillatory solutions which sometimes even cause the loss of hyperbolicity. We consider the stochastic Galerkin scheme, in which we filter the coefficients of the polynomial expansion in order to reduce oscillations. We further apply the multi-element approach and ensure the preservation of hyperbolic solutions through the hyperbolicity limiter. In addition to that, we study the intrusive polynomial moment method, which guarantees hyperboli…
We present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor [22] back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent so…
A continuous/discontinuous Galerkin solution of the shallow water equations with dynamic viscosity, high-order wetting and drying, and implicit time integration
Stanford University（斯坦福大学）；Naval Postgraduate School（海军研究生院）；Argonne National Laboratory（阿贡国家实验室）；The University of Chicago（芝加哥大学）
This study presents a two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC). Firstly, the experimental order of convergence of the scheme is shown to see the second-order accuracy in time. Secondly, the effect of the TBC on a simple domain is discussed; the artificial reflections are kept from the Dirichlet boundaries and removed significantly from the transmission boundaries. Thirdly, the scheme is applied to a complex practical domain, i.e., the Bay of Bengal region, which is non-convex and includes islands. The effect of the TBC is di…
Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation
The discontinuous Galerkin finite element method (DGFEM) developed by Rhebergen et al. (2008) offers a robust method for solving systems of nonconservative hyperbolic partial differential equations but, as we show here, does not satisfactorily deal with topography in shallow water flows at lowest order (so-called DG0, or equivalently finite volume). In particular, numerical solutions of the space-DG0 discretised one-dimensional shallow water equations over varying topography are not truly `well-balanced'. A numerical scheme is well-balanced if trivial steady states are satisfied in the numeric…
This paper aims to simulate viscoplastic flow in a shallow-water regime. We specifically use the Bingham model in which the material behaves as a solid if the stress is below a certain threshold, otherwise, it moves as a fluid. The main difficulty of this problem is the coupling of the shallow-water equations with the viscoplastic constitutive laws and the high computational effort needed in its solution. Although there have been many studies of this problem, most of these works use explicit methods with simplified empirical models. In our work, to accommodate non-uniform grids and complicated…
The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or th…
In this paper, we investigate steady states of shallow water moment equations including bottom topographies. We derive a new hyperbolic shallow water moment model based on linearized moment equations that allows for a simple assessment of the steady states. After proving hyperbolicity of the new model, the steady states are fully identified. A well-balanced scheme is adopted to the specific structure of the new model and allows to preserve the steady states in numerical simulations.
The Savage-Hutter (SH) equations are a hyperbolic system of nonlinear partial differential equations describing the temporal evolution of the depth and depth averaged velocity for modelling the avalanche of a shallow layer of granular materials on an inclined surface. These equations admit the occurrence of shock waves and vacuum fronts as in the shallow-water equations while possessing the special reposing state of granular material. In this paper, we develop a third-order Runge-Kutta discontinuous Galerkin (RKDG) method for the numerical solution of the one-dimensional SH equations. We adopt…
Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction
We develop and analyze a discontinuous Petrov--Galerkin method with optimal test functions (DPG method) for a shallow shell model of Koiter type. It is based on a uniformly stable ultraweak formulation and thus converges robustly quasi-uniformly. Numerical experiments for various cases, including the Scordelis--Lo cylindrical roof, elliptic and hyperbolic geometries, illustrate its performance. The built-in DPG error estimator gives rise to adaptive mesh refinements that are capable to resolve boundary and interior layers. The membrane locking is dealt with by raising the polynomial degree onl…
Penalty-scaling effects in nonsymmetric interior-penalty DG discretizations of viscous rotating shallow-water equations
College of Science, National University of Defense Technology（国防科技大学理学院）；College of Meteorology and Oceanography, National University of Defense Technology（国防科技大学气象海洋学院）
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
INSA Toulouse（图卢兹国立应用科学学院）；University of Montpellier, Institut Montpelliérain Alexander Grothendieck and Inria LEMON team（蒙彼利埃大学，亚历山大·格罗滕迪克蒙彼利埃研究所和Inria LEMON团队）
Numerical climate- and weather-prediction requires the fast solution of the equations of fluid dynamics. Discontinuous Galerkin (DG) discretisations have several advantageous properties. They can be used for arbitrary domains and support a structured data layout, which is important on modern chip architectures. For smooth solutions, higher order approximations can be particularly efficient since errors decrease exponentially in the polynomial degree. Due to the wide separation of timescales in atmospheric dynamics, semi-implicit time integrators are highly efficient, since the implicit treatme…
