A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids
Introduction
The discontinuous Galerkin methods [1], [2] (DGM) have recently become popular for the solution of systems of conservation laws to arbitrary order of accuracy. The discontinuous Galerkin methods combine two advantageous features commonly associated to finite element and finite volume methods. As in classical finite element methods, accuracy is obtained by means of high-order polynomial approximation within an element rather than by wide stencils as in the case of finite volume methods. The physics of wave propagation is, however, accounted for by solving the Riemann problems that arise from the discontinuous representation of the solution at element interfaces. In this respect, the methods are therefore similar to finite volume methods. In fact, the basic cell-centered finite volume scheme exactly corresponds to the DG(0) method, i.e., to the discontinuous Galerkin method using piecewise constant polynomials. Consequently, the DG(p) method with can be regarded as the natural extension of finite volume methods to higher-order methods. The discontinuous Galerkin methods have many distinguished features: (1) The methods are well suited for complex geometries since they can be applied on unstructured grids. In addition, the methods can also handle non-conforming elements, where the grids are allowed to have hanging nodes. (2) The methods are compact, as each element is independent. Since the elements are discontinuous, and the inter-element communications are minimal (elements only communicate with von Neumann neighbors regardless of the order of accuracy of the scheme), they are highly parallelizable. The compactness also allows for structured and simplified coding for the methods. (3) They can easily handle adaptive strategies, since refining or coarsening a grid can be achieved without considering the continuity restriction commonly associated with the conforming elements. The methods allow easy implementation of hp-refinement, for example, the order of accuracy, or shape, can vary from element to element. (4) They have several useful mathematical properties with respect to conservation, stability, and convergence.
However, the discontinuous Galerkin methods have a number of their own weaknesses. In particular, how to effectively control spurious oscillations in the presence of strong discontinuities remains one of unresolved issues in the DG methods. Like any higher-order schemes (>1), the discontinuous Galerkin methods will suffer from non-physical oscillations in the vicinity of discontinuities that exist in problems governed by hyperbolic conservation laws. Two common approaches to address this issue are a discontinuity capturing and an appropriate slope limiter. The former adds explicitly consistent artificial viscosity terms to the discontinuous Galerkin discretization. The main disadvantage of this approach is that it usually requires some user-defined parameters, which can be both mesh and problem dependent. Classical techniques of slope limiting are not directly applicable for high-order DGM because of the presence of volume terms in the formulation. Therefore, the slope limiter is not integrated in the computation of the residual, but effectively acts as a post-processing filter. Many slope limiters used in the finite volume method (FVM) can then be used or modified to meet the needs of the DGM. Unfortunately, the use of the limiter will reduce the order of accuracy to first order in the presence of discontinuities. Furthermore, the active limiters in the smooth extrema will pollute the solution in the flow field and ultimately destroy the higher-order accuracy of DGM [3]. Indeed, the limiters used in TVD/MUSCL finite volume methods are less robust than the strategies of essential non-oscillatory (ENO) and weighted ENO (WENO) finite volume methods. The ENO schemes were initially introduced by Harten et al. [4] in which oscillations up to the order of the truncation error are allowed to overcome the drawbacks and limitations of limiter-based schemes. ENO schemes avoid interpolation across high-gradient regions through biasing of the reconstruction. This biasing is achieved by reconstructing the solution on several stencils at each location and selecting the reconstruction which is in some sense the smoothest. This allows ENO schemes to retain higher-order accuracy near high-gradient regions. However, the selection process can lead to convergence problems and loss of accuracy in regions with smooth solution variations. To counter these problems, the so-called weighted ENO scheme introduced by Liu et al. [5] is designed to present better convergence rate for steady state problems, better smoothing for the flux vectors, and better accuracy using the same stencils than the ENO scheme. WENO scheme uses a suitably weighted combination of all reconstructions rather than just the one which is judged to be the smoothest. The weighting is designed to favor the smooth reconstruction in the sense that its weight is small, if the oscillation of a reconstructed polynomial is high and its weight is order of one, if a reconstructed polynomial has low oscillation. Qiu and Shu initiated the use of WENO scheme as limiters for the DG method [6] for solving 1D and 2D Euler equations on structured grids. Later on, they constructed a class of WENO schemes based on Hermite polynomials, termed as HWENO (Hermite WENO) schemes and applied this HWENO as limiters for the DG methods [7], [8]. The main difference between HWENO and WENO schemes is that the former has a more compact stencil than the latter for the same order of accuracy.
Unfortunately, implementation of both ENO and WENO schemes is fairly complicated on arbitrary meshes, especially in 3D. In fact, there are very few results obtained using ENO/WENO on unstructured grids in 3D especially for higher-order reconstruction. Harten and Chakravarthy [9], Abgrall [10], and Sonar [11] presented the first implementation of ENO schemes on unstructured triangular grids. Implementations of WENO methods on unstructured triangular grids were also presented by Friedrich [12] and Hu and Shu [13].
In the present work, a WENO reconstruction scheme based on the Hermite polynomials is presented and used as a non-linear limiter for a discontinuous Galerkin method to solve the compressible Euler equations on unstructured grids. The new reconstruction scheme makes use of the invaluable information, namely derivatives that are handily available in the context of the discontinuous Galerkin method, thus making the implementation of WENO schemes straightforward on unstructured grids in both 2D and 3D. Only the van Neumann neighborhood is required for the construction of stencils, regardless of the order of solution polynomials to be reconstructed. The resulting HWENO reconstruction keeps full conservation of mass, momentum, and energy, is uniformly accurate with no overshoots and undershoots, is easy to implement on arbitrary meshes, has good convergence properties, and is computationally efficient. This HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter. The remainder of this paper is structured as follows. The governing equations are listed in Section 2. The underlying discontinuous Galerkin method is presented in Section 3. The construction and implementation of the limiter based on the HWENO scheme are described in detail in Section 4. Extensive numerical experiments are reported in Section 5. Concluding remarks are given in Section 6.
Section snippets
Governing equations
The Euler equations governing unsteady compressible inviscid flows can be expressed in conservative form aswhere the conservative state vector U and the inviscid flux vectors F are defined bywhere the summation convention has been used and ρ, p, and e denote the density, pressure, and specific total energy of the fluid, respectively, and ui is the velocity of the flow in the coordinate direction xi. This set of equations is completed
Discontinuous Galerkin spatial discretization
To formulate the discontinuous Galerkin method, we first introduce the following weak formulation of Eq. (2.1), which is obtained by multiplying Eq. (2.1) by a test function W, integrating over the domain Ω, and performing an integration by parts:where denotes the boundary of Ω, and the unit outward normal vector to the boundary.
Assuming that Ωh is a classical triangulation of Ω where the domain Ω is subdivided into a collection of non-overlapping
Hermite WENO reconstruction
The DG method described above will produce non-physical oscillations and even nonlinear instability for flows with strong discontinuities. A common solution to this problem is to use a slope limiter. Unfortunately, DGM are very sensitive to the treatment and implementation of the slope limiters [3]. Slope limiters frequently identify regions near smooth extrema as requiring limiting, and this typically results in a reduction of the optimal high-order convergence rate. For aerodynamic
Numerical examples
All computations are performed on a Dell Precision M70 laptop computer with 2GBytes memory running the Suse 10.0 Linux operating system. The explicit three-stage third-order TVD Runge–Kutta scheme is used for unsteady flow computations and the p-multigrid for steady-state flow problems. For steady-state solutions, the relative L2 norm of the density residual is taken as a criterion to test convergence history. All calculations are started with uniform flow. An elaborate and well-tested finite
Concluding remarks
A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin finite element method on unstructured grids. The HWENO reconstruction algorithm uses the invaluable derivative information, that are handily available in the context of discontinuous Galerkin methods, thus only requires the van Neumann neighborhood for the construction of stencils. This significantly facilitates the implementation of WENO
Acknowledgments
The first author expresses appreciation to Prof. Shu at Brown University and Prof. Cockburn at University of Minnesota for many helpful, instructive, and fruitful discussions about DG and WENO.
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