Florian Hindenlang

Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors

Abstract We consider a family of explicit one-step time discretizations for finite volume and discontinuous Galerkin schemes, which is based on a predictor-corrector formulation. The predictor remains local taking into account the time evolution of the data only within the grid cell. Based on a space–time Taylor expansion, this idea is already inherent in the MUSCL finite volume scheme to get second order accuracy in time and was generalized in the context of higher order ENO finite volume schemes. We interpret the space–time Taylor expansion used in this approach as a local predictor and conclude that other space–time approximate solutions of the local Cauchy problem in the grid cell may be applied. Three possibilities are considered in this paper: (1) the classical space–time Taylor expansion, in which time derivatives are obtained from known space-derivatives by the Cauchy–Kovalewsky procedure; (2) a local continuous extension Runge–Kutta scheme and (3) a local space–time Galerkin predictor with a version suitable for stiff source terms. The advantage of the predictor–corrector formulation is that the time evolution is done in one step which establishes optimal locality during the whole time step. This time discretization scheme can be used within all schemes which are based on a piecewise continuous approximation as finite volume schemes, discontinuous Galerkin schemes or the recently proposed reconstructed discontinuous Galerkin or P N P M schemes. The implementation of these approaches is described, advantages and disadvantages of different predictors are discussed and numerical results are shown.

Mesh Curving Techniques for High Order Discontinuous Galerkin Simulations

In the development of the next generation of numerical methods for CFD, Discontinuous Galerkin methods are promising a substantial increase in efficiency and accuracy. While the particular high order methods can be very distinct, they have in common that they must rely on a high-order approximation of curved geometries to maintain their high-order of accuracy. The generation of curved meshes is thus a topic whose importance cannot be overstated, if one truly wants to apply DG methods to problems with industrial relevance. Especially aerospace applications heavily rely on complex geometries and pose high requirements to the quality of geometry representation. In this work we present several techniques to produce high order meshes, relying on linear meshes, which can be generated by standard commercial mesh generation tools. We describe the generation of the curved surface meshes and curved volume meshes, which can be particularly difficult for curved boundary layers.

A RUNGE-KUTTA BASED DISCONTINUOUS GALERKIN METHOD WITH TIME ACCURATE LOCAL TIME STEPPING

An explicit one-step time discretization for discontinuous Galerkin schemes applied to advection-diffusion equations is presented. The main idea is based on a predictor corrector approach which was proposed in [1]. The interesting feature is that the predictor is local and takes only into account the time evolution of the data within each grid cell. In this paper, we focus ourselves to a continuous extension Runge-Kutta schemes which was described in [2]. The advantage of the predictor corrector formulation is that the time evolution is done in one step and the data of the direct neighbors are needed only. Hence, the proposed discontinuous Galerkin scheme has the optimal locality within the whole time step. This is the basis to introduce a timeconsistent local time stepping in a way such that every grid cell may run with its own optimal time step as given by the local stability restriction [3]. The time accuracy and the efficiency of the local time stepping is shown for linear and non-linear problems. A direct numerical simulation of the aeroacoustics of a natural gas injector shows the efficiency of the presented methodology, being well suited for unsteady advection dominated problems with adaptive schemes.

The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations

In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267–279, 1997) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier–Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835–B867, 2014) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.

An Efficient High Performance Parallelization of a Discontinuous Galerkin Spectral Element Method

We describe an efficient parallelization strategy for the discontinuous Galerkin spectral element method, illustrated by a structured grid framework. Target applications are large scale DNS and LES calculations on massively parallel systems. Due to the simple and efficient formulation of the method, a parallelization aiming at one-element-per-processor calculations is feasible; a highly desired feature for emerging multi- and many-core architectures. We show scale-up tests on up to 131,000 processors.

Discontinuous Galerkin for High Performance Computational Fluid Dynamics (hpcdg)

In this paper we present selected ongoing computations, performed on HLRS clusters. Three efficient explicit Discontinuous Galerkin schemes, suitable for high performance calculations, are employed to perform direct numerical simulations of isotropic turbulence and turbulent channel flow, large eddy simulations of cavity-flows as well as hybrid simulations of aeroacoustic phenomena. The computations were performed on hundreds to thousands computer cores.

Discontinuous Galerkin Schemes for the Direct Numerical Simulation of Fluid Flow and Acoustics

The direct numerical simulation of uid ow by large eddy simulation with noise generation and propagation needs a very exible numerical approach to be e cient. We show that the class of discontinuous Galerkin schemes has a high potential to satisfy all these requirements. We describe several building blocks of our development in the last years to apply these schemes to the direct aeroacoustic simulations: A fast spectral element DG scheme for Cartesian and hexahedron grids, resolution of turbulent ow and acoustic with high order schemes, their e cient use on massively parallel computer systems as well as the direct visualization of piecewise high order polynomial data.

Highly Efficient and Scalable Software for the Simulation of Turbulent Flows in Complex Geometries

This paper investigates the efficiency of simulations for compressible turbulent flows with noise generation in complex geometries. It analyzes two different approaches and their suitability with respect to quality as well as turn around times required in industrial DoE processes. One approach makes use of a high order discontinuous Galerkin scheme. The efficiency of high order schemes on coarser meshes is compared to lower order schemes on finer meshes. The second approach is a 2nd order Finite Volume scheme, which employs a zonal coupling of LES and RANS to enhance efficiency in turbulence simulation. The schemes are applied to three industrial test cases which are described. Difficulties on HPC systems, especially load-balancing, MPI and IO, are pointed out and solutions are presented.

Numerical Investigation of High-Order Gyrotron Mode Propagation in Launchers at 170 GHz

This paper presents for the first time the transient simulation of the 3-D mode converter with surface deformation. The simulation results were obtained by solving the Maxwell equations directly on a 3-D domain with a discontinuous Galerkin method. The presented full-wave simulation was possible only through the use of an advanced highly scalable numerical method operating on a large-scale high-performance computing system. Moreover, the properties of the Maxwell solver with respect to accuracy and computation time are discussed for the application to a smooth-wall waveguide where an analytical solution is available.

Unstructured three-dimensional High Order Grids for Discontinuous Galerkin Schemes

Discontinuous Galerkin (DG) methods are a prominent candidate for high order accurate schemes for advection dominated problems in threedimensional complex geometries, since they sustain high order spatial accuracy even on general unstructured grids. To maintain the high order accuracy at curved wall boundaries, a high order representation of the elements near the wall surface is required, i.e. high order grids. Regarding three-dimensional curved geometries, the construction of such unstructured curved grids is a subtle task. The implementation of the DG scheme will be discussed, focusing on curved element mappings. The mappings of curved element sides as well as the curved element volume are described. We present our approach of the high order grid generation process in detail. The main idea is to rely on established unstructured grid generators for a basic volume grid consisting of straight sided elements and provide additionally high order information for the curved boundaries. We analyze the properties of the geometry approximation with a simple test case using the Maxwell equations. Finally, we present an unstructured curved mesh of a fairly complex aircraft, showing the applicability of the approach.