Solving PDEs (partial differential equations) is one of the most important fundamental tasks for solving physics and engineering problems. Efficient numerical algorithms for existing and future computer architectures are one of the main targets. CERFACS performs research on the mathematical fundamentals and on application cases that are of specific interest for CERFACS shareholders, and for which the methods are developed and tested.
Research and progress in PDE methods is deeply rooted in functional analysis, but often the usefulness in practice will depend on the availability of an efficient implementation, and, in particular, the interface to numerical solvers from linear algebra, both direct and iterative. In some applications, unconventional methods, such as mesh-free and kinetic methods (such as the lattice Boltzmann method, LBM) and semistructured mesh methods combined with matrix-free techniques have shown superior potential for high performance computing. In terms of the resolution achieved and the efficiency of execution, we have demonstrated that such methods can outperform conventional techniques by several orders of magnitude. This gain in efficiency promises enormous savings or, equivalently, a superior fidelity and improved predictive power of the simulations.
A – Legacy codes: CERFACS has successfully developed solvers for PDEs. This is the case of the LES reacting flow solver AVBP. Preparing such codes for future architectures is a continuing task at CERFACS. A related field of development for these legacy codes is fully adaptive meshing techniques to hide all meshing tasks for end users.
B – Future methods and solvers: Research in numerical PDEs is progressing rapidly. CERFACS explores promising paths for future PDE solvers. A special example is the LBM which is already studied intensively at CERFACS in coordination with leading laboratories in the field (M2P2 in Marseille, FAU Erlangen). The LBM can be implemented efficiently on certain architectures and is especially suited for manycore accelerators. In this field, the ALGO and CFD teams work together. In addition to the LBM approach, work has been carried out on another equally promising approach: SDM (Spectral Difference Method which is a family of high-oder discontinuous approach). The latter makes it possible to obtain very accurate LES simulations (up to now spatial order up to ten have been tested) with a competitive cost compared to traditional approaches. CERFACS developed its own platform called JAGUAR and since 2018 CERFACS shares the ownership and the development of this solver with ONERA.