Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis

The use of high‐order polynomials in discontinuous Galerkin (DG) approximations to convection‐dominated transport problems tends to cause a violation of the maximum principle in regions where the derivatives of the solution are large. In this paper, we express the DG solution in terms of Taylor basis functions associated with the cell average and derivatives at the center of the cell. To control the (derivatives of the) discontinuous solution, the values at the vertices of each element are required to be bounded by the means. This constraint is enforced using a hierarchical vertex‐based slope limiter to constrain the coefficients of the Taylor polynomial in a conservative manner starting with the highest‐order terms. The loss of accuracy at smooth extrema is avoided by taking the maximum of the correction factors for derivatives of order p and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. In the case of a non‐orthogonal Taylor basis, the same limiter is applied to the vector of discretized time derivatives before the multiplication by the off‐diagonal part of the consistent mass matrix. This strategy leads to a remarkable gain of accuracy, especially in the case of simplex meshes. A numerical study is performed for a 2D convection equation discretized with linear and quadratic finite elements. Copyright © 2012 John Wiley & Sons, Ltd.     

An improvement of classical slope limiters for high‐order discontinuous Galerkin method

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.     

适用于间断Galerkin方法的限制器研究

开发了一种适用于高精度间断Galerkin方法的斜率（多项式系数）限制器。与现有的斜率限制器不同，该限制器实施过程不考虑网格单元类型（三角形或四边形），通过全微分方法构造新的多项式系数，因此，该限制器能够适用于各种类型网格——结构化网格、具有单一单元的非结构化网格和具有混合单元的非结构化网格。由于该限制器能够方便地应用于具有混合单元的非结构化网格，因此，本文使用的程序能够方便地求解具有复杂几何结构的流动问题。本文利用一些典型算例对其性能进行了验证，表明该限制器适用于不同类型的网格单元，能够在光滑解区保证高的精度，并能够在阊断区抑帛3非物理振荡。     

A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third‐order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.     

Anisotropic mesh refinement for discontinuous Galerkin methods in two‐dimensional aerodynamic flow simulations

We derive and implement two types of anisotropic indicators which can be used within an anisotropic refinement algorithm for second but also for higher‐order discontinuous Galerkin discretizations. Although the first type of indicator employs the possible inter‐element discontinuities of the discrete functions, the second type of indicator estimates the approximation error in terms of second but possibly also higher‐order derivatives. We implement a simple extension of these indicators to systems of equations which performs similar to the so‐called metric intersection used to combine the metric information of several solution components and is applicable to higher‐order discretizations as well. The anisotropic indicators are incorporated into an adaptive refinement algorithm which uses state‐of‐the‐art residual‐based or adjoint‐based indicators for goal‐oriented refinement to select the elements to be refined, whereas the anisotropic indicators determine which anisotropic case the selected elements shall be refined with. We demonstrate the performance of the anisotropic refinement algorithm for sub‐, trans‐ and supersonic, inviscid and viscous compressible flows around a NACA0012 airfoil. Copyright © 2007 John Wiley & Sons, Ltd.     

A numerical analysis of a class of generalized Boussinesq‐type equations using continuous/discontinuous FEM

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ²ⁿ⁺²) term (n ∈ ?) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P₂ Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes

In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell‐centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell‐centered unknowns, which reduces the scheme to a completely cell‐centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity‐preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity‐preserving cell‐centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Copyright © 2010 John Wiley & Sons, Ltd.     

An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive mesh refinement for high‐resolution finite element schemes

New a posteriori error indicators based on edgewise slope‐limiting are presented. The L₂‐norm is employed to measure the error of the solution gradient in both global and element sense. A second‐order Newton–Cotes formula is utilized in order to decompose the local gradient error from a ??₁ finite element solution into a sum of edge contributions. The slope values at edge midpoints are interpolated from the two adjacent vertices. Traditional techniques to recover (superconvergent) nodal gradient values from consistent finite element slopes are reviewed. The deficiencies of standard smoothing procedures—L₂‐projection and the Zienkiewicz–Zhu patch recovery—as applied to nonsmooth solutions are illustrated for simple academic configurations. The recovered gradient values are corrected by applying a slope limiter edge‐by‐edge so as to satisfy geometric constraints. The direct computation of slopes at edge midpoints by means of limited averaging of adjacent gradient values is proposed as an inexpensive alternative. Numerical tests for various solution profiles in one and two space dimensions are presented to demonstrate the potential of this postprocessing procedure as an error indicator. Finally, it is used to perform adaptive mesh refinement for compressible inviscid flow simulations. Copyright © 2006 John Wiley & Sons, Ltd.     

One‐dimensional shock‐capturing for high‐order discontinuous Galerkin methods

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd.