The 2015 AMMCS-CAIMS Congress

Interdisciplinary AMMCS Conference Series

Waterloo, Ontario, Canada | June 7-12, 2015

AMMCS-CAIMS 2015 Plenary Talk

Recent progress in the development of parameter free continuous finite element methods for compressible fluids

Rémi Abgrall (University of Zurich)

In this talk, I will review the current status of the so-called Residual Distribution schemes applied, in particular, to compressible fluid dynamics problems. Other physical models include the Shallow Water equation and generalization, MHD, etc.

After the early work of R. Ni at Bombardier, and the seminal work of P.L Roe, in particular his 1981 JCP paper and its extensions to scalar multidimensional schemes, these schemes can be considered as finite element methods of the streamline diffusion type. The emphasis is put on non-oscillatory properties, in order to be able to compute flow discontinuities, so that they are nonlinear by construction. Indeed shock capturing is done in a totally different manner as for streamline diffusion, allowing for a class of parameter free schemes. In a way, the Residual Distribution methods can be seen as a kind of compromise between high order TVD-like finite difference/finite volume schemes and classical finite element methods, in that they borrow ideas from both communities: geometrical flexibility, the residual concept on one side, and non oscillatory, maximum principle on the other one.

In the talk, we will first consider the case of steady scalar hyperbolic problems, showing how one can systematically construct parameter free essentially non-oscillatory schemes. Then we will move towards steady advection diffusion problems, showing how uniform accuracy, whatever the Peclet/Reynolds number is. The last part of the talk we will consider recent work on unsteady problems. Examples of compressible flows (laminar and turbulent) will be also shown, in order to demonstrate the efficiency of the method, both in accuracy, memory footprint and CPU time.

This is joint work with many colleagues and students among whom Dante de Santis, Mario Ricchiuto, Algiane Froehly, Adam Larat, Mohamed Mezine at INRIA, and many discussions with H. Deconinck (VKI, Belgium) as well as Phil Roe (Michigan, USA). This work has been funded by several EU contracts: the FP6 ADIGMA project (contract AST5-CT-2006-030719), the FP7 IDIHOM project (contract AAT-2010-RTD-1-265780) and the ERC Advanced Grant ADDECCO (contract #226316), as well as a grant of the Swiss National Fund.

Rémi Abgrall is a former student of Ecole Normale Supérieure de Saint Cloud. After his PhD, he has been an engineer at ONERA, then a research scientist at INRIA. Since January 2014, he is professor at the University of Zürich, Institute of Mathematics, after having been Professor in the University of Bordeaux (Institut Polytechnique de Bordeaux) since 1996 and in secondment at INRIA from 2008 till the end of 2013. He has been awarded an Advanced Research Grant from the ERC in December 2008 and has been an invited speaker at the International Conference of Mathematicians (ICM 2014) in Seoul. He is associate editor of several international journals, including the Journal of Computational Physics, Mathematics of Computation, Computers and Fluids, and the Journal of Scientific Computing. He is also co-chief editor of the International Journal on Numerical Methods in Fluids. His research is about efficient algorithms for the simulation of compressible materials (single fluids, multiphase, interface problems, compressible solids) using high order schemes designed for unstructured meshes. He also has interest in (curved) meshes generation for high order scheme and model reduction for transport dominated problems with and without discontinuities in the solution.