By revisiting the basic Godunov approach for system of linear hyperbolic Partial Differential Equations (PDEs) we show that it is hybridizable. As such, it is a natural recipe for us to ...constructively and systematically establish a unified hybridized discontinuous Galerkin (HDG) framework for a large class of PDEs including those of Friedrichs' type. The unification is fourfold. First, it provides a single constructive procedure to devise HDG schemes for elliptic, parabolic, hyperbolic, and mixed-type PDEs. The key that we exploit is the fact that, for many PDEs, irrespective of their type, the first order form is a hyperbolic system. Second, it reveals the nature of the trace unknowns as the upwind states. Third, it provides a parameter-free HDG framework, and hence eliminating the “usual complaint” that HDG is a parameter-dependent method. Fourth, it allows us to rediscover most existing HDG methods and furthermore discover new ones.
We apply the proposed unified framework to three different PDEs: the convection–diffusion–reaction equation, the Maxwell equation in frequency domain, and the Stokes equation. The purpose is to present a step-by-step construction of various HDG methods, including the most economic ones with least trace unknowns, by exploiting the particular structure of the underlying PDEs. The well-posedness of the resulting HDG schemes, i.e. the existence and uniqueness of the HDG solutions, is proved. The well-posedness result is also extended and proved for abstract Friedrichs' systems. We also discuss variants of the proposed unified framework and extend them to the popular Lax–Friedrichs flux and to nonlinear PDEs. Numerical results for transport equation, convection–diffusion equation, compressible Euler equation, and shallow water equation are presented to support the unification framework.
We present a class of high-order finite element methods that can conserve the linear and angular momenta as well as the energy for the equations of linear elastodynamics. These methods are devised by ...exploiting and preserving the Hamiltonian structure of the equations of linear elastodynamics. We show that several mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin (HDG) methods belong to this class. We discretize the semidiscrete Hamiltonian system in time by using a symplectic integrator in order to ensure the symplectic properties of the resulting methods, which are called symplectic Hamiltonian finite element methods. For a particular semidiscrete HDG method, we obtain optimal error estimates and present, for the symplectic Hamiltonian HDG method, numerical experiments that confirm its optimal orders of convergence for all variables as well as its conservation properties.
•The first Hamiltonian structure-preserving HDG method for linear elastodynamics.•A complete analysis of energy-conserving Mixed, DG and HDG methods.•The linear and angular momenta, and the total energy remain constant in time.•HDG methods for linear elastodynamics with optimal error estimates.
We present a systematic and constructive methodology to devise various hybridized discontinuous Galerkin (HDG) methods for linearized shallow water equations. It is shown that using the ...Rankine--Hugoniot condition to solve the Riemann problem is a natural approach to deriving HDG methods. At the heart of our development is an upwind HDG framework obtained by hybridizing the upwind flux in the standard discontinuous Galerkin (DG) approach. Essentially, the HDG framework is a redesign of the standard DG approach to reducing the number of coupled unknowns. An upwind and three other HDG methods are constructed and analyzed for linearized shallow water systems. Rigorous stability and convergence analysis for both semidiscrete and fully discrete systems are provided. We extend the upwind HDG method to a family of penalty HDG schemes and rigorously analyze their well-posedness, stability, and convergence rates. Numerical results for the linear standing wave and the Kelvin wave for oceanic shallow water systems a...
In this work, we present HODG, an open-source component-based development framework based on high order Discontinuous Galerkin (DG) methods for solving compressible Euler and Navier-Stokes equations. ...This framework is written in pure C++11, and proposes “component” as the basic function unit, which is the key to the Interface-Oriented Programming principle and Aspect-Oriented Programming (AOP) technology. Built on the top-level design of components, HODG is a flexible yet pragmatic development framework that works right out of the box and is easy to use for starters and developers.
The current release of HODG supports structured, unstructured, hybrid, and second-order meshes available. It is capable of solving Euler and Navier-Stokes (N-S) equations in two dimensions and three dimensions. In this development framework, spatial accuracy is easy to be extended to higher orders, and DG-P0, DG-P1, DG-P2 (up to third order) are implemented in the current version. Moreover, implicit temporal discretization and explicit discretization are available. Especially for viscous fluxes, the direct Discontinuous Galerkin (DDG) formulation and Bassi and Rebay II (BR2) scheme are implemented. In the case of strong discontinuities or shock waves, artificial viscosity is applied to capture shock waves. Various benchmark numerical examples are provided to demonstrate the full capabilities of HODG. The software is freely available under an MIT license.
Program Title: HODG v2.0.0
CPC Library link to program files:https://doi.org/10.17632/835732yz8s.1
Code Ocean capsule:https://codeocean.com/capsule/4788938
Licensing provisions: MIT
Programming language: C++11
Nature of problem: Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.
Solution method: DG-P0, DG-P1, DG-P2 spatial discretization, combining with DDG and BR2 schemes for viscous fluxes, explicit and implicit time schemes, and supporting structured, unstructured, hybrid, and high-order meshes.
unusual features: developed based on modern C++11 features; propose the “component” as basic function unit, which follows the practice of Interface-Oriented Programming principle; Special component named “action” is used, and is the key to AOP technology in this framework; Implicitly implements the Dependency Injection (DI) technology; Flexible yet pragmatic, and works right out of the box.
In this paper, we develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and ...the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique, where the numerical fluxes in the DG scheme are treated separately. However, if the numerical approximations are close to the steady state, the numerical fluxes are not independent, and it is possible to use the relationship to obtain a new penalty parameter which is zero at the steady state and the full scheme is PP. To be more precise, we first reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions with zero velocities. Numerical experiments are given to show the performance of the proposed methods.
•Construct positivity-preserving and well-balanced schemes for Euler equations with gravitation.•The penalty is chosen to be zero in the Lax-Friedrichs fluxes at the steady-states.•A novel positivity-preserving technique is developed.•The scheme is straightforward to implement.
In this work, we propose an Exponential DG framework for partial differential equations. We decompose 7 governing equations into linear and nonlinear parts to which we apply the discontinuous ...Galerkin 8 (DG) spatial discretization. In particular, we construct the linear part using Jacobian that effectively 9 capture stiff characteristics in the system. The former is integrated analytically, whereas the latter 10 is approximated. This approach i) is stable with a large Courant number (Cr > 1); ii) supports 11 high-order solutions both in time and space; iii) is computationally favorable compared to IMEX 12 DG methods with no preconditioner; iv) becomes comparable to explicit RKDG methods on uniform 13 mesh and beneficial on non-uniform grid for Euler equations; v) is scalable in a modern massively 14 parallel computing architecture due to its explicit nature of exponential time integrators and com15 pact communication stencil of DG method. Numerical results demonstrate the performance of our 16 proposed methods through various examples. We also discuss the stability and convergence analysis 17 for our exponential DG scheme in the context of Burgers equation.
In this paper, we investigate the convergence behavior of discontinuous Galerkin methods for solving a class of delay differential equations. Although discontinuities may occur in various orders of ...the derivative of the solutions, we show that the m-degree DG solutions have (m+1)th order accuracy in L∞ norm. Numerical experiments confirm the theoretical results of the methods.
Summary
To develop a robust, well‐balanced and quadrature‐free Runge‐Kutta discontinuous Galerkin (RKDG) shallow water solver, we introduce an efficient wetting and drying (WD) treatment in this ...paper. The main feature of this WD treatment is the use of vertex‐based linear reconstructed solutions in transition (partially wet) regions and high‐order solutions in smooth wet areas. To preserve the positivity of water depth, we also propose a modified time step size with the quadrature‐free scheme. The advantages of the WD treatment include robustness and the capability to address arbitrary high‐order RKDG methods in wet regions, thus making the quadrature‐free scheme more accurate and applicable to various flooding and drying problems. Two numerical test cases are used to validate the numerical method, and the results indicate that the WD method is accurate and robust for different flow regimes with dry areas.
We propose an efficient and robust p‐adaptive wetting and drying treatment for the well‐balanced RKDG method in the shallow water modeling. The new model is able to preserve the well‐balanced property and ensure the positivity of water depth with arbitrary high‐order basis functions.
In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow ...in evolving geometries. The framework uses implicitly defined meshes—wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries—and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement.
In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid–structure interaction, and free surface flow. A class of techniques known as interfacial gauge methods is adopted to solve the corresponding incompressible Navier–Stokes equations, which, compared to archetypical projection methods, have a weaker coupling between fluid velocity, pressure, and interface position, and allow high-order accurate numerical methods to be developed more easily. Convergence analyses conducted throughout the work demonstrate high-order accuracy in the maximum norm for all of the applications considered; for example, fourth-order spatial accuracy in fluid velocity, pressure, and interface location is demonstrated for surface tension-driven two phase flow in 2D and 3D. Specific application examples include: vortex shedding in nontrivial geometry, capillary wave dynamics revealing fine-scale flow features, falling rigid bodies tumbling in unsteady flow, and free surface flow over a submersed obstacle, as well as high Reynolds number soap bubble oscillation dynamics and vortex shedding induced by a type of Plateau–Rayleigh instability in water ripple free surface flow. These last two examples compare numerical results with experimental data and serve as an additional means of validation; they also reveal physical phenomena not visible in the experiments, highlight how small-scale interfacial features develop and affect macroscopic dynamics, and demonstrate the wide range of spatial scales often at play in interfacial fluid flow.
We present several high-order accurate finite element methods for the Maxwell’s equations which provide time-invariant, non-drifting approximations to the total electric and magnetic charges, and to ...the total energy. We devise these methods by taking advantage of the Hamiltonian structures of the Maxwell’s equations as follows. First, we introduce spatial discretizations of the Maxwell’s equations using mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin methods to obtain a semi-discrete system of equations which display discrete versions of the Hamiltonian structure of the Maxwell’s equations. Then we discretize the resulting semi-discrete system in time by using a symplectic integrator. This ensures the conservation properties of the fully discrete system of equations. For the Symplectic Hamiltonian HDG method, we present numerical experiments which confirm its optimal orders of convergence for all variables and its conservation properties for the total linear and angular momenta, as well as the total energy. Finally, we discuss the extension of our results to other boundary conditions and to numerical schemes defined by different weak formulations.
•A new HDG method for the electric field and magnetic vector potential formulation of Maxwell’s equations.•The first symplectic Hamiltonian hybridizable discontinuous Galerkin method for Maxwell’s equations.•Energy-conserving Mixed, Discontinuous Galerkin and Hybridizable Discontinuous Galerkin methods.•A comprehensive analysis of the Hamiltonian structures of Mixed, DG, and HDG methods for Maxwell’s equations.