Geometric multigrid with implicit relaxation schemes for Finite Volume and Discontinuous Galerkin discretizations of compressible flows

Kurzfassung

The accurate, reliable and efficient solution of the Navier-Stokes equations for complex industrial applications is an ongoing challenge in the field of computational fluid dynamics (CFD). In the last decade higher order methods have become an attractive alternative to the established finite volume methods. In particular, discontinuous Galerkin (DG) methods are very attractive due to the fact that they can be viewed alternatively as an extended finite volume method or as finite element technique. However, solution strategies for DG methods are so far inferior when compared with those established for finite volume methods. Several approaches use an explicit (pseudo-)time-stepping based on a Runge-Kutta method (RKDG), but the associated effort becomes prohibitive for high resolution computations on finer meshes. Efficient solution methods for DG methods are usually implemented considering a stabilized Newton or implicit Euler method requiring the exact derivative of the residual function, but this typically yields ill-conditioned linear systems that are hard to solve. Convergence can be accelerated using a so-called p-multigrid, in which the multigrid idea is established via reducing the order of the method while keeping the same computational mesh. Unfortunately, experience has shown that this kind of multigrid is rather limited and does not show the same desired effects as a geometric multigrid, in particular if applied to stiff problems arising from highly stretched meshes, high Reynolds numbers and turbulence models. On the other hand, also for second order finite volume methods for unstructured meshes, where an agglomerated multigrid in combination with a rediscretization on the coarse meshes is used, often a loss of reliability of the multigrid algorithm can be observed, in particular when the agglomerated meshes become very coarse. However, with a reduced depth of the coarse meshes the convergence rate achieved via the multigrid procedure degenerates more or less to a single-grid iteration. Recently, the rediscretization process has been exchanged by a Galerkin type projection method. With this modification the reliability for an agglomerated multigrid in the context of finite volume methods could be significantly improved, such that even for turbulent flows with high Reynolds number multigrid algorithms with more or less arbitrary cycle depth can be applied successfully and the full power of multigrid can be exploited. Guided by the wish to incorporate also a geometric multigrid component for DG methods, the idea of a modified multigrid was successfully carried over to higher order methods. Similar to the fact, that the agglomerated multigrid by Galerkin projection uses a discretization only on the finest mesh, also for DG methods the evaluation of the discrete flow equations is only required on the finest mesh and therefore the quadrature rules necessary for a DG code are well defined also on the coarse meshes. This is usually not the case if a discretization is performed directly on the coarse meshes which consist of polyhedral elements. As DG methods employ upwind schemes where the high order function representation is achieved through individual unknowns, not via reconstruction, a rediscretization on the coarse levels with an integration over the fine mesh elements and inter-element faces is equivalent to a Galerkin projection of the fine mesh equations for convective terms and thus for the Euler equations. Remaining differences between DG and finite volume methods may be found in the formulation of the implicit relaxation scheme in the multigrid algorithm. As the order of the scheme does not influence the number of unknowns in the finite volume case, an approximate derivative usually corresponding to a first order discretization can be used as smoother. This is not possible in the DG context, where the full derivative is required. However, the use of an exact linearization can be expected to yield improved convergence rates, which might help to compensate the additional computational complexity. We will show for both the finite volume and the DG method numerical results which highlight the significant improvements achieved by a modified agglomerated multigrid using only fine-level evaluations of the discretization. We demonstrate on some basic test cases that the design orders of both methods are reached. Moreover, in the DG case we demonstrate improvements when compared with the established p-multigrid ansatz and furthermore highlight that curvilinear meshes are necessary in order to reach the design order of the schemes. On standard straight-sided meshes the accuracy of DG methods even falls behind unstructured second order finite volume methods on the same mesh. Furthermore, convergence when analyzing the entropy error versus the equivalent mesh size. Furthermore, the nomalized timings of the solution process show a nearly linear scaling with the number of unknowns.