Robert Klöfkorn

A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE

In a companion paper (Bastian et al. 2007, this issue) we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structures. Thus, user implementations become independent of the underlying grid implementation. Modern C++ template techniques are used to provide an interface implementation without big performance losses. The implementation is realized as part of the software environment DUNE (http://dune-project.org/). Numerical tests demonstrate the flexibility and the efficiency of our approach.

A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework

We give a mathematically rigorous definition of a grid for algorithms solving partial differential equations. Unlike previous approaches (Benger 2005, PhD thesis; Berti 2000, PhD thesis), our grids have a hierarchical structure. This makes them suitable for geometric multigrid algorithms and hierarchical local grid refinement. The description is also general enough to include geometrically non-conforming grids. The definitions in this article serve as the basis for an implementation of an abstract grid interface as C++ classes in the framework (Bastian et al. 2008, this issue).

Towards a Unified Framework for Scientific Computing

Most finite element, or finite volume software is built around a fixed mesh data structure. Therefore, each software package can only be used efficiently for a relatively narrow class of applications. For example, implementations supporting unstructured meshes allow the approximation of complex geometries but are in general much slower and require more memory than implementations using structured meshes. In this paper we show how a generic mesh interface can be defined such that one algorithm, e. g. a discretization scheme, works on different mesh implementations. For a cell centered finite volume scheme we show that the same algorithm runs thirty times faster on a structured mesh implementation than on an unstructured mesh and is only four times slower than a non-generic version for a structured mesh. The generic mesh interface is realized within the Distributed Unified Numerics Environment DUNE.

A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems

In this paper we present a stabilized Discontinuous Galerkin (DG) method for hyperbolic and convection dominated problems. The presented scheme can be used in several space dimension and with a wide range of grid types. The stabilization method preserves the locality of the DG method and therefore allows to apply the same parallelization techniques used for the underlying DG method. As an example problem we consider the Euler equations of gas dynamics for an ideal gas. We demonstrate the stability and accuracy of our method through the detailed study of several test cases in two space dimension on both unstructured and cartesian grids. We show that our stabilization approach preserves the advantages of the DG method in regions where stabilization is not necessary. Furthermore, we give an outlook to adaptive and parallel calculations in 3d.

Compact and Stable Discontinuous Galerkin Methods for Convection-Diffusion Problems

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 results 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

An efficient implementation of an adaptive and parallel grid in DUNE

L^2

A Bottom-up approach to Grid-Computing at a University: the Black-Forest-Grid Initiative

-accuracy and CPU time.

On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture

In this contribution we describe and evaluate an efficient implementation of an adaptive and parallel grid (ALUGrid) within the Distributed and Unified Numerics Environment DUNE. A generalization of the serial grid interface of DUNE, described in [1], to the adaptive and parallel case is discussed and example computations using the grid interface are presented. The computations are compared with computations of the original code, which was optimized for the specific example problem studied here.

Efficient Parallel Simulation of Atherosclerotic Plaque Formation Using Higher Order Discontinuous Galerkin Schemes

ABSTRACT Recent years have seen a rapid increase in the need for highperformance computing. These demands come from disciplines such as particle physics traditionally relying on High Performance Computing (HPC) but lately also from the various branches of life science that have matured into quantitative disciplines. The classical infrastructure of university computer centres results to be unsuited to cope with the new requirements for a multitude of reasons. Here we discuss the causes of this failure and present a solution developed at the University of Freiburg in a collaborative effort of several faculties. We demonstrate that using state of the art grid computing technology the problem can now be addressed in a bottom-up approach. The organizational, technical, and financial components of our framework, the Black Forest Grid Initiative (BFG) are described and results of its implementation are presented. In the process, a number of new questions have emerged which the next phase of our project needs to address.

Performance Pitfalls in the Dune Grid Interface

We present and discuss discontinuous Galerkin (DG) schemes for dry and moist atmospheric flows in the mesoscale. We derive terrain-following coordinates on the sphere in strong-conservation form, which makes it possible to perform the computation on a Cartesian grid and yet conserves the momentum density on an f -plane. A new DG model, i.e. DG-COSMO, is compared to the operational model COSMO of the Deutscher Wetterdienst (DWD). A simplified version of the suggested terrain-following coordinates is implemented in DG-COSMO and is compared against the DG dynamical core implemented within the DUNE framework, which uses unstructured grids to capture orography. Finally, a few idealised test cases, including 3d and moisture, are used for validation. In addition an estimate of efficiency for locally adaptive grids is derived for locally and non-locally occurring phenomena.