A high-order velocity-based discontinuous Galerkin scheme for the shallow water equations: local conservation, entropy stability, well-balanced property, and positivity preservation
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 solution unknown, no specific numerical quadrature rules are needed to achieve entropy stability of our scheme on general unstructured meshes in two dimensions.
The proposed DG semi-discretization is then carefully combined with the classical explicit strong stability preserving Runge-Kutta (SSP-RK) time integrators [13] to yield a locally conservative, well-balanced, and positivity preserving fully discrete scheme. Here the positivity preservation property is enforced with the help of a simple scaling limiter. In the fully discrete scheme, we re-introduce discharge as an auxiliary unknown variable. In doing so, standard slope limiting procedures can be applied on the conservative variables (water height and discharge) without violating the local conservation property. Here we apply a characteristic-wise TVB limiter [5] on the conservative variables using the Fu-Shu troubled cell indicator [10] in each inner stage of the Runge-Kutta time stepping to suppress numerical oscillations.
High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes…
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…
The under integration of the volume terms in the discontinuous Galerkin spectral element approximation introduces errors at non-conforming element faces that do not cancel and lead to free-stream preservation errors. We derive volume and face conditions on the geometry under which a constant state is preserved. From those, we catalog six special cases on the geometry that preserve a constant state, the most general being to approximate the geometry sub-parametrically to one half the order of the solution. Numerical examples are presented to illustrate the results.
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semi-discretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary $θ$-method in time (including the forward and backward Euler methods, and a second-order method by choosing $θ\in [0,1]$ suitably). The full discretization generates a two-pa…
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 present consistent and inconsistent discontinuous Galerkin methods for incompressible Euler and Navier-Stokes equations with the kinematic pressure, Bernoulli function and EMAC function. Semi- and fully discrete energy stability of the proposed dG methods are proved in a unified fashion. Conservation of total energy, linear and angular momentum is discussed with both central and upwind fluxes. Numerical experiments are presented to demonstrate our findings and compare our schemes with conventional schemes in the literature in both unsteady and steady problems. Numerical resul…
Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part II: The multidimensional case
High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable DG methods which incorporate an "entropy projection" are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.
A number of key scientific computing applications that are based upon tensor-product grid constructions, such as numerical weather prediction (NWP) and combustion simulations, require property-preserving interpolation. Essentially non-oscillatory (ENO) interpolation is a classic example of such interpolation schemes. In the aforementioned application areas, property preservation often manifests itself as a requirement for either data boundedness or positivity preservation. For example, in NWP, one may have to interpolate between the grid on which the dynamics is calculated to a grid on which t…
A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
University of Wisconsin–Platteville（威斯康星大学普拉特维尔分校）；Instituto Tecnológico Autónomo de México（墨西哥自治理工学院）；Iowa State University（爱荷华州立大学）；Minnesota State University Moorhead（明尼苏达摩尔黑德州立大学）；University of Florida（佛罗里达大学）
The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor-corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear…
Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural soluti…
This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux.…
In this paper, we present a positivity-preserving limiter for nodal Discontinuous Galerkin disctretizations of the compressible Euler equations. We use a Legendre-Gauss-Lobatto (LGL) Discontinuous Galerkin Spectral Element Method (DGSEM) and blend it locally with a consistent LGL-subcell Finite Volume (FV) discretization using a hybrid FV/DGSEM scheme that was recently proposed for entropy stable shock capturing. We show that our strategy is able to ensure robust simulations with positive density and pressure when using the standard and the split-form DGSEM. Furthermore, we show the applicabil…
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Mathematical Institute, Faculty of Mathematics and Physics, Charles University（布拉格查理大学数理学院数学研究所）；Department of Mathematics, Baylor University（贝勒大学数学系）；Department of Mathematics and Statistics, Memorial University of Newfoundland（纽芬兰纪念大学数学与统计系）
In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving properties, such as positivity preservation and entropy inequality hold. We demonstrate how to ensure them and prove the convergence of our multidimensional high-order DG scheme via dissipative weak solutions. In numerical simulations, we verify our theoretical results.
This paper designs and analyzes positivity-preserving well-balanced (WB) central discontinuous Galerkin (CDG) schemes for the Euler equations with gravity. A distinctive feature of these schemes is that they not only are WB for a general known stationary hydrostatic solution, but also can preserve the positivity of the fluid density and pressure. The standard CDG method does not possess this feature, while directly applying some existing WB techniques to the CDG framework may not accommodate the positivity and keep other important properties at the same time. In order to obtain the WB and posi…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Eötvös Loránd University（厄特沃什·罗兰大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part I: The one-dimensional case
In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third o…
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…
Nodal discontinuous Galerkin methods for non-ideal equations of state: pressure equilibrium preservation and entropy correction
Oden Institute for Computational Engineering and Sciences, The University of Texas-Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Department of Aerospace Engineering & Engineering Mechanics, The University of Texas-Austin（德克萨斯大学奥斯汀分校航空航天工程与应用力学系）；Institute of Mathematics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学数学研究所）；Department of Mathematics, University of Hamburg（汉堡大学数学系）；Laboratories for Computational Physics and Fluid Dynamics, U.S. Naval Research Laboratory（美国海军研究实验室计算物理与流体动力学实验室）；Amentum — U.S. Air Force Research Laboratory (Edwards)（阿门特姆——美国空军研究实验室（爱德华兹））
We consider the periodic initial-value problem for the Serre equations of water-wave theory and its semidiscrete approximation in the space of smooth periodic polynomial splines. We prove that the semidiscrete problem is well posed, locally in time, and satisfies a discrete positivity property for the water depth.
We propose an explicit, single step discontinuous Galerkin (DG) method on moving grids using the arbitrary Lagrangian-Eulerian (ALE) approach for one dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Se…
A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ens…
In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy are positive, and the mass fraction of each species is between 0 and 1. Secondly, due to the rapid reaction rate, the system may contain stiff sources, and the strong-stability-preserving explicit Runge-Kutta method may result in limited time step sizes. To obtain physically relevant numerical approximations, we apply the bound-preserving technique to the DG methods. F…
