Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case

A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional non-linear hyperbolic conservation law systems, was developed and applied as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods in [J. Comput. Phys. 193 (2003) 115]. In this paper, we extend the method to solve two dimensional non-linear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.     

Fourier Type Error Analysis of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

In this paper we present Fourier type error analysis on the recent four discontinuous Galerkin methods for diffusion equations, namely the direct discontinuous Galerkin (DDG) method (Liu and Yan in SIAM J. Numer. Anal. 47(1):475?C698, 2009); the DDG method with interface corrections (Liu and Yan in Commun. Comput. Phys. 8(3):541?C564, 2010); and the DDG method with symmetric structure (Vidden and Yan in SIAM J. Numer. Anal., 2011); and a DG method with nonsymmetric structure (Yan, A discontinuous Galerkin method for nonlinear diffusion problems with nonsymmetric structure, 2011). The Fourier type L ² error analysis demonstrates the optimal convergence of the four DG methods with suitable numerical fluxes. The theoretical predicted errors agree well with the numerical results.     

Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for?Conservation Laws on Triangular Meshes

In Zhang and Shu (J. Comput. Phys. 229:3091–3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918–8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.     

Implicit space–time residual distribution method for unsteady laminar viscous flow

The class of multidimensional upwind residual distribution (RD) schemes has been developed in the past decades as an attractive alternative to the finite volume (FV) and finite element (FE) approaches. Although they have shown superior performances in the simulation of steady two-dimensional and three-dimensional inviscid and viscous flows, their extension to the simulation of unsteady flow fields is still a topic of intense research [ICCFD2, International Conference on Computational Fluid Dynamics 2, Sydney, Australia, 15–19 July 2002; M. Mezine, R. Abgrall, Upwind multidimensional residual schemes for steady and unsteady flows].Recently the space–time RD approach has been developed by several researchers [Int. J. Numer. Methods Fluids 40 (2002) 573; J. Comput. Phys. 188 (2003) 16; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002; J. Comput. Phys. 188 (2003) 16; R. Abgrall; M. Mezine, Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems] which allows to perform second order accurate unsteady inviscid computations. In this paper we follow the work done in [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002]. In this approach the space–time domain is discretized and solved as a (d+1)-dimensional problem, where d is the number of space dimensions. In [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002] it is shown that thanks to the multidimensional upwinding of the RD method, the solution of the unsteady problem can be decoupled into sub-problems on space–time slabs composed of simplicial elements, allowing to obtain a true time marching procedure. Moreover, the method is implicit and unconditionally stable for arbitrary large time-steps if positive RD schemes are employed.We present further development of the space–time approach of [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002] by extending it to laminar viscous flow computations. A Petrov–Galerkin treatment of the viscous terms [Project Report 2002-06, von Karman Institute for Fluid Dynamics, Belgium, 2002; J. Dobeš, Implicit space–time method for laminar viscous flow], consistent with the space–time formulation has been investigated, implemented and tested. Second order accuracy in both space and time was observed on unstructured triangulation of the spatial domain.The solution is obtained at each time-step by solving an implicit non-linear system of equations. Here, following [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002], we formulate the solution of this system as a steady state problem in a pseudo-time variable. We discuss the efficiency of an explicit Euler forward pseudo-time integrator compared to the implicit Euler. When applied to viscous computation, the implicit method has shown speed-ups of more than a factor 50 in terms of computational time.     

Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method,III: Unstructured Meshes

In [J. Comput. Phys. 193:115–135, 2004] and [Comput. Fluids 34:642–663, 2005], Qiu and Shu developed a class of high order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving nonlinear hyperbolic conservation law systems, and applied them as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods on structured meshes. In this continuation paper, we extend the method to solve two dimensional problems on unstructured meshes. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods. The research was partially supported by the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simulations, NSFC grant 10671091 and JSNSF BK2006511.     

Solutions of Multi-dimensional Hyperbolic Systems of Conservation Laws by Square Entropy Condition Satisfying Discontinuous Galerkin Method

In this paper, we study formally high-order accurate discontinuous Galerkin methods on general arbitrary grid for multi-dimensional hyperbolic systems of conservation laws [Cockburn, B., and Shu, C.-W. (1989, Math. Comput. 52, 411–435, 1998, J. Comput. Phys. 141, 199–224); Cockburn et al. (1989, J. Comput. Phys. 84, 90–113; 1990, Math. Comput. 54, 545–581). We extend the notion of E-flux [Osher (1985) SIAM J. Numer. Anal. 22, 947–961] from scalar to system, and found that after flux splitting upwind flux [Cockburn et al. (1989) J. Comput. Phys. 84, 90–113] is a Riemann solver free E-flux for systems. Therefore, we are able to show that the discontinuous Galerkin methods satisfy a cell entropy inequality for square entropy (in semidiscrete sense) if the multi-dimensional systems are symmetric. Similar result [Jiang and Shu (1994) Math. Comput. 62, 531–538] was obtained for scalar equations in multi-dimensions. We also developed a second-order finite difference version of the discontinuous Galerkin methods. Numerical experiments have been obtained with excellent results.      

A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for Hamilton-Jacobi equations. The L ² stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.     

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P_NP_M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P_NP_M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].     

On improving mass-conservation properties of the hybrid particle-level-set method

For the computation of multi-phase flows level-set methods are an attractive alternative to volume-of-fluid or front-tracking approaches. For improving their accuracy and efficiency the hybrid particle-level-set modification was proposed by Enright et al. [Enright D, Fedkiw R, Ferziger J, Mitchell I. A hybrid particle-level-set method for improved interface capturing. J Comput Phys 2002;183:83-116]. In actual applications the overall properties of a level-set method, such as mass conservation, are strongly affected by discretization schemes and algorithmic details. In this paper we address these issues with the objective of determining the optimum alternatives for the purpose of direct numerical simulation of dispersed-droplet flows. We evaluate different discretization schemes for curvature and unit normal vector at the interface. Another issue is the particular formulation of the reinitialization of the level-set function which significantly affects the quality of computational results. Different approaches employing higher-order schemes for discretization, supplemented either by a correction step using marker particles (Enright et al., 2002) or by additional constraints [Sussman M, Almgren AS, Bell JB, Colella P, Howell LH, Welcome ML. An adaptive level set approach for incompressible two-phase flows. J Comput Phys 1999;148:81-124] are analyzed. Different parameter choices for the hybrid particle-level-set method are evaluated with the purpose of increasing the efficiency of the method. Aiming at large-scale computations we find that in comparison with pure level-set methods the hybrid particle-level-set method exhibits better mass-conservation properties, especially in the case of marginally resolved interfaces.     

Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.This revised version was published online in July 2005 with corrected volume and issue numbers.     

An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows

This paper presents the latest developments of a discontinuous Galerkin (DG) method for incompressible flows introduced in [Bassi F, Crivellini A, Di Pietro DA, Rebay S. An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J Comput Phys 2006;218(2):794–815] for the steady Navier–Stokes equations and extended in [Bassi F, Crivellini A. A high-order discontinuous Galerkin method for natural convection problems. In: Wesseling P, Oñate E, Periaux J, editors. Electronic proceedings of the ECCOMAS CFD 2006 conference, Egmond aan Zee, The Netherlands, September 5–8; 2006. TU Delft] to the coupled Navier–Stokes and energy equations governing natural convection flows.

The method is fully implicit and applies to the governing equations in primitive variable form. Its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of local Riemann problems associated with the artificial compressibility perturbation of the Euler equations. The tight coupling between pressure and velocity so introduced stabilizes the method and allows using equal-order approximation spaces for both pressure and velocity. Since, independently of the amount of artificial compressibility added, the interface flux reduces to the physical one for vanishing interface jumps, the resulting method is strongly consistent.

In this paper, we present a review of the method together with two recently developed issues: (i) the high-order DG discretization of the incompressible Euler equations; (ii) the high-order implicit time integration of unsteady flows. The accuracy and versatility of the method are demonstrated by a suite of computations of steady and unsteady, inviscid and viscous incompressible flows.     

Direct Discontinuous Galerkin Method and Its Variations for Second Order Elliptic Equations

In this paper, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under \(L^2\) norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal \((k+1)\)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal \((k+1)\)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.     

Selecting the Numerical Flux in Discontinuous Galerkin Methods for Diffusion Problems

In this paper we present numerical investigations of four different formulations of the discontinuous Galerkin method for diffusion problems. Our focus is to determine, through numerical experimentation, practical guidelines as to which numerical flux choice should be used when applying discontinuous Galerkin methods to such problems. We examine first an inconsistent and weakly unstable scheme analyzed in Zhang and Shu, Math. Models Meth. Appl. Sci. (M ³ AS) 13, 395–413 (2003), and then proceed to examine three consistent and stable schemes: the Bassi–Rebay scheme (J. Comput. Phys. 131, 267 (1997)), the local discontinuous Galerkin scheme (SIAM J. Numer. Anal. 35, 2440–2463 (1998)) and the Baumann–Oden scheme (Comput. Math. Appl. Mech. Eng. 175, 311–341 (1999)). For an one-dimensional model problem, we examine the stencil width, h-convergence properties, p-convergence properties, eigenspectra and system conditioning when different flux choices are applied. We also examine the ramifications of adding stabilization to these schemes. We conclude by providing the pros and cons of the different flux choices based upon our numerical experiments.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.     

Computation of incompressible bubble dynamics with a stabilized finite element level set method

This paper presents a stabilized finite element method for the three dimensional computation of incompressible bubble dynamics using a level set method. The interface between the two phases is resolved using the level set approach developed by Sethian [Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999], Sussman et al. [J. Comput. Phys. 114 (1994) 146], and Sussman et al. [J. Comput. Phys. 148 (1999) 81–124]. In this approach the interface is represented as a zero level set of a smooth function. The streamline-upwind/Petrov–Galerkin method was used to discretize the governing flow and level set equations. The continuum surface force (CSF) model proposed by Brackbill et al. [J. Comput. Phys. 100 (1992) 335–354] was applied in order to account for surface tension effects. To restrict the interface from moving while re-distancing, an improved re-distancing scheme proposed in the finite difference context [J. Comput. Phys. 148 (1999) 81–124] is adapted for finite element discretization. This enables us to accurately compute the flows with large density and viscosity differences, as well as surface tension. The capability of the resultant algorithm is demonstrated with two and three dimensional numerical examples of a single bubble rising through a quiescent liquid, and two bubble coalescence.     

On different flux splittings and flux functions in WENO schemes for balance laws

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].     

Simulation of a viscous compressible flow past a circular cylinder with high-order discontinuous Galerkin methods

This paper is devoted to the simulation of viscous compressible flows with high-order accurate discontinuous Galerkin methods on bidimensional unstructured meshes. The formulation for the solution of the Navier–Stokes equations is due to Oden et al. [An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, The University of Texas at Austin, 1997; J Comput Phys 1998;146:491–519]. It involves a weak imposition of continuity conditions on the state variables and on fluxes across interelement boundaries. It does not make use of any auxiliary variables and then the cost for the implementation is reasonable. The method is coupled with a limiting procedure recently developed by the authors to suppress oscillations near large gradients. The limiter is totally free of problem dependence and maintains the convergence order for errors measured in the L¹-norm. This paper presents detailed numerical results of a viscous compressible flow past a circular cylinder at a Reynolds number of 100 for the cases of subsonic and supersonic regimes. The proposed simulations suggest that the method is very robust and is able to produce very accurate results on unstructured meshes.