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 results show that global conservation of the physical quantities may not be enough to demonstrate the performance of the schemes, and our schemes are competitive and able to capture essential physical features in several benchmark problems.
We present a class of Arbitrary Lagrangian-Eulerian hybridizable discontinuous Galerkin methods for the incompressible flow with moving boundaries and interfaces including two-phase flow with surface tension.
A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier-Stokes equations is formulated and analyzed. We prove unconditional stability of the propose scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates.
A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach global systems with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the s…
This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first order time extrapolation which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well-posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order.
We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier--Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is solved by an H1-conforming finite element method, and an upwind discontinuous Galerkin finite element method with post-processed velocity is adopted for the density equation. The proposed method is proved to be convergent in approximating reasonably smooth solutions in three-dimensional convex polyhedral domains.
This paper proposes a fully implicit numerical scheme for immiscible incompressible two-phase flow in porous media taking into account gravity, capillary effects, and heterogeneity. The objective is to develop a fully implicit stable discontinuous Galerkin (DG) solver for this system that is accurate, bound-preserving, and locally mass conservative. To achieve this, we augment our DG formulation with post-processing flux and slope limiters. The proposed framework is applied to several benchmark problems and the discrete solutions are shown to be accurate, to satisfy the maximum principle and l…
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.
Wormhole propagation plays a very important role in the product enhancement of oil and gas reservoir. A new combined hybrid mixed finite element method is proposed to solve incompressible wormhole propagation problem with discontinuous Galerkin finite element procedure, in which, the new hybrid mixed finite element algorithm is established for pressure equation, while the discontinuous Galerkin finite element method is considered for concentration equation, and then the porosity function is computed straightly by the approximate value of the concentration. This new combined method can keep loc…
In this paper, we propose a unified numerical framework for the time-dependent incompressible Navier--Stokes equation which yields the $H^1$-, $H(\text{div})$-conforming, and discontinuous Galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness. Under minimum assumption on Galerkin spaces, the semi- and fully-discrete stability is proved when a family of implicit Runge--Kutta methods are used for time discretization. Furthermore, we present a unified discussion on the penalty term. Numerical experiments are presented to compare our schemes wi…
Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, { in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in $L^2$-norm in spatial direction,} which however is not optimal and valid only under certain extra restrictions on both time ste…
Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here …
In this contribution we present a local discontinuous Galerkin (LDG) pressure-correction scheme for the incompressible Navier-Stokes equations. The scheme does not need penalty parameters and satisfies the discrete continuity equation exactly. The scheme is especially suitable for two-phase flow when used with a piecewise-linear interface construction (PLIC) volume-of-fluid (VoF) method and cut-cell quadratures.
We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace solution fields, we arrive at an energy stable method which returns pointwise divergence-free velocity fields and magnetic fields and properly balances linear momentum. We discretize in time using a second-order-in-time generalized-$α$ method, and we present a block iterative method for solving the resulting nonlinear system of equations at each time step. We nume…
We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation…
We present a reduced order method (ROM) based on proper orthogonal decomposition (POD) for the viscous Burgers' equation and the incompressible Navier-Stokes equations discretized using an implicit-explicit hybrid discontinuous Galerkin/discoutinuous Galerkin (IMEX HDG/DG) scheme. A novel closure model, which can be easily computed offline, is introduced. Numerical results are presented to test the proposed POD model and the closure model.
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 high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting
University of Delaware（特拉华大学）；Texas Tech University（德克萨斯理工大学）
The accurate numerical simulation of high Reynolds number incompressible flows is a challenging topic in computational fluid dynamics. Classical inf-sup stable methods like the Taylor-Hood element or only $L^2$-conforming discontinuous Galerkin (DG) methods relax the divergence constraint in the variational formulation. However, unlike divergence-free methods, this relaxation leads to a pressure-dependent contribution in the velocity error which is proportional to the inverse of the viscosity, thus resulting in methods that lack pressure robustness and have difficulties in preserving structure…
In this work a solver for instationary two-phase flows on the basis of the extended Discontinuous Galerkin (extended DG/XDG) method is presented. The XDG method adapts the approximation space conformal to the position of the interface. This allows a sub-cell accurate representation of the incompressible Navier-Stokes equations in their sharp interface formulation. The interface is described as the zero set of a signed-distance level-set function and discretized by a standard DG method. For the interface, resp. level-set, evolution an extension velocity field is used and a two-staged algorithm …
In this paper we consider a conservative discretization of the two-dimensional incompressible Navier--Stokes equations. We propose an extension of Arakawa's classical finite difference scheme for fluid flow in the vorticity-stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.
A hybridizable discontinuous Galerkin (HDG) formulation of the linearized incompressible Navier-Stokes equations, known as Oseen equations, is presented. The Cauchy stress formulation is considered and the symmetry of the stress tensor and the mixed variable, namely the scaled strain-rate tensor, is enforced pointwise via Voigt notation. Using equal-order polynomial approximations of degree k for all variables, HDG provides a stable discretization. Moreover, owing to Voigt notation, optimal convergence of order k+1 is obtained for velocity, pressure and strain-rate tensor and a local postproce…
In this work we introduce the development of a three--phase incompressible Navier--Stokes/Cahn--Hilliard numerical method to simulate three--phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three--phase model adopted in this work is a Cahn--Hilliard diffuse interface model, which was derived by Boyer and Lapuerta et al. 2006. The Cahn--Hilliard model is coupled to the entropy--stable incompressible Navier--Stokes equations model derived by Manzanero et al. …
In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the disctinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment. Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport…
We develop an entropy stable two-phase incompressible Navier--Stokes/Cahn--Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn-Hilliard equation as the phase field method, a skew-symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free- and no-slip wall boundary conditions with non-zero wall contact angle. We then construct a high-order DG approximation of the model that satisfies the SBP-SAT property. With the help of a discrete stability anal…
A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of d…
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…
Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part II
Institute for Analysis and Scientific Computing, TU Wien（维也纳工业大学分析与科学计算研究所）；Institute for Numerical and Applied Mathematics, University of Göttingen（哥廷根大学数值与应用数学研究所）
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, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method.
