Compact and Stable Discontinuous Galerkin Methods for Convection-Diffusion Problems

S. Brdar, A. Dedner, R. Klöfkorn
2012 SIAM Journal on Scientific Computing  
We present a new scheme, the compact discontinuous Galerkin 2 (CDG2) method, for solving nonlinear convection-diffusion problems together with a detailed comparison to other well-accepted DG methods. The new CDG2 method is similar to the CDG method that was recently introduced in the work of Perraire and Persson for elliptic problems. One main feature of the CDG2 method is the compactness of the stencil which includes only neighboring elements, even for higher order approximation. Theoretical
more » ... sults showing coercivity and stability of CDG2 and CDG for the Poisson and the heat equation are given, providing computable bounds on any free parameters in the scheme. In numerical tests for an elliptic problem, a scalar convection-diffusion equation, and for the compressible Navier-Stokes equations, we demonstrate that the CDG2 method slightly outperforms similar methods in terms of L 2 -accuracy and CPU time. order requires only information from direct neighbors. This is a key feature for efficient computation on today's multicore parallel architectures. Various versions of the DG method to solve elliptic problems have emerged over the years, and it is interesting to mention the work [2] which unifies several of them in an abstract framework and provides analysis of their accuracy and stability for Poisson's equation. Two of the methods mentioned in [2] which satisfy the properties stated above are the interior penalty (IP) (introduced in [18]) or Bassi-Rebay 2 (BR2) (introduced in [4, 5]). More recently the compact discontinuous Galerkin (CDG) method was introduced in [36]. The IP scheme with different stabilization terms is analyzed for the two-dimensional (2D) compressible nonlinear Navier-Stokes equations in the work of Hartmann and Houston [27] . Additional stabilization of the IP method is based on the penalization of jumps of the numerical solution across grid interfaces which has to take into account the order of the method. Estimates of the penalization parameters for one-dimensional parabolic problems are presented in [34] , for 2D elliptic problems in [1, 19, 20] , and for 2D compressible nonlinear Navier-Stokes equations in [26] . In [22, 38] the problem of estimating the penalty coefficient, in case of simplified diffusion term, is transformed into a problem of finding an estimate for a series of inequalities between different norms. The BR2 method is compared with IP in [27] (see also the references therein). The stabilization of the BR2 method, as well as for the CDG and the CDG2 methods, is based on special lifting operators. This approach may come at a considerable computational cost, since the lifting operators need to be computed on both grid elements which share an interface. The advantage of CDG and CDG2 over BR2 is exactly at this point, as they require the evaluation of one lifting operator on only one side of each interface. In the context of nonlinear problems this is even more important since one might consider a matrix free implementation of these methods. Calculations of these liftings add a nonnegligible part to the computational cost of the scheme especially on general quadrilateral and hexahedral grids. The rest of this paper is organized as follows. In section 2, we describe the CDG2, CDG, BR2, and IP methods in a suitable form for the stability analysis carried out in section 3. Here, the analysis of the coercivity in the case of Poisson's equations and L 2 -stability in the case of a linear heat equation is carried out for CDG and CDG2. In section 4, we highlight implementation details. Most notably we use the stability estimate for the CDG2 method to derive a special switching function which improved the performance of the method considerably. Practical results, including comparisons of the new CDG2 method with CDG, IP, and BR2, are presented in section 5. Conclusions are drawn in section 6. DG formulation for convection-diffusion equations. In this section we will derive the primal DG formulation for general nonlinear convection-diffusionreaction equations of the form and Ω ⊂ R d is a bounded subset with polygonal (for d = 2, or polyhedral for d = 3) boundary. In this paper we focus, in particular, on the discretization of the diffusion term in (2.1). To this end we first consider the discretization of a linear elliptic problem Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. COMPACT AND STABLE DG METHODS A265 with variable coefficients which will serve as a building block for the discretization of (2.1) discussed in section 2.2. The discretization is described for a 2D problem, but it is straightforward to extend it to three space dimensions. Elliptic problems. In order to derive the discretization of the diffusion term in (2.1) we consider the following elliptic problem in R d , d = 2, of the form
doi:10.1137/100817528 fatcat:t2b5bv6yn5g2fekxiadojmqvue