Developing accurate schemes depends highly on the mesh chosen for the computations. Even if classical approaches (implicit compact, Finite Difference, WENO... schemes) are easy to implement in a structured framework, the situation is much more complex for unstructured grids. In particular, any approach with large stencil must be avoided for several reasons. First, computing large stencil needs complex geometric algorithms to implement, validate and specific situations can be encountered, which may lead to implement correction(s). In particular, such a situation generally occurs when the mesh is composed of elements with different shapes. Moreover, large stencil and High Performance Computing (HPC) are two topics with opposite objectives. It is clear that many fields must be exchanged when a large stencil is considered and therefore messages have large length. The sequential efficiency is also impacted and if data are badly arranged in memory, cache misses lead to a high CPU time per degree of freedom. Finally, the efficiency of such codes is rather connected with memory bandwidth and MPI efficiency (of the supercomputer) than with the kernel itself. But unstructured grids are unavoidable for computations over complex CAD and a lot of work has been published during the last years on high order schemes for unstructured grids.
A way to overcome large stencils on unstructured grids lies on increasing the number of degrees of freedom inside the element. It is convenient to define high order representations of quantities inside any mesh element following a polynomial approximation. The reconstructed variables at the faces depend on the mesh element only and two different extrapolations (one at left side and one at right one) may lead to discontinuous flow at mesh faces. Among the methods proposed in the literature, we have identified three techniques:
- The Discontinuous Galerkin -DG- technique is based on the Finite
Element framework. The principle is to look for a polynomial
representation of the solution that satisfies a variational form of
the governing system within each element. Even if the technique is
quite old (Reed and Hill - 1973), its extension to the full Navier
Stokes equations is recent and many papers have been published during
the last 10 years.
- The Spectral Volume -SV- technique is based on the Finite Volume
framework and it follows the pioneering work of Wang (2002). It
consists in defining element subdivisions on which a
classical Finite Volume technique is considered. The mean quantity over
each volume is necessary to build the high order representation of data
inside the element.
- The Spectral Difference -SD- technique follows the Finite Difference approach. Kopriva and Kolias published it in 1996 and a more general presentation of the technique was published in 2006 by Liu, Vinokur and Wang. The idea is to define high order approximation of the quantities from Finite Differences inside each mesh cell.
DEPICT Prototype
For the AAM team, Hugues Deniau, Jean-François Boussuge and myself have initiated the development of a new prototype platform designed to test high order discretization techniques on unstructured grids. We have decided to focus our attention on the Spectral Difference method. This choice is motivated by several reasons: the SD method has been built in order to correct some drawbacks of DG and SV. First, it seems more efficient in term of CPU usage (less computations per degree of freedom) than DG technique. Moreover, SV suffers a high sensitivity with respect to element decomposition and this drawback is avoided with SD method. Finally, the SD approach is less mature and the potential of research work is greater.
The SD method is currently implemented in a development prototype called DEPICT (DEvelopment Prototype for hIgh order schemes on unstruCTured grids). It is written in fortran 90, with a (quite) simple data structure. Parallel computations can be performed with a MPI implementation and asynchronous communications. Finally, a second order classical Finite Volume has also been implemented in DEPICT in order to tackle comparisons with the classical FV technique.
List of contributors
N. Villedieu (Post-doc), G. Puigt, P. Cayot (PhD student), M. Kuzmin (training student), H. Deniau