In this presentation, we address the objective of accurately simulating multi-scale PDEs in time and space at a reasonable cost. This leads to the use of dynamic mesh adaptation and dedicated numerical schemes. Our main focus for this seminar is on examining the impact of mesh adaptation on the accuracy of the solution. We draw attention to the potential coupling of mesh adaptation and numerical scheme for the classical approaches and explain why we intend to use fully adaptive multi-resolution in the sense of Harten based on wavelets. We present a theory based on modified equations for the convective part of the PDEs and include the fully adaptive multi-resolution in the error estimate. It provides a theoretical basis for understanding the impact of mesh adaptation / mesh compression on the quality of results. We will also briefly present the open-source code framework we use "samurai", as well as a new data-structure based on interval representation and algebra of sets. This data-structure allows to both reach a very small memory trace of the compressed mesh as well as handle all the potential versions of the AMR and adaptive multiresolution with a single tool. The use of another library "ponio" allows to connect the mesh adaptation library to various time integration strategies.
Biography
Marc Massot obtained his Ph.D. in Applied Mathematics from École polytechnique in 1996. After a year at Yale University, Department of Mechanical Engineering, he obtained a CNRS position in the Applied Mathematics Laboratory of the University of Lyon, where he stayed until 2005. He was offered an Associate Professor position at École Centrale Paris when he installed a mathematics team in the EM2C mechanical engineering laboratory. He was Visiting Professor at the Center for Turbulence Research at Stanford University in 2011-2012 and created and chaired the Fédération de Mathématiques de l’École Centrale Paris between 2013 and 2016. He initiated the Computing Center (Mésocentre) of École Centrale Paris in 2010, of which he was the deputy director until 2016 and he is scientific adviser at ONERA since 2013. Full Professor since 2011, he has been recruited on a full Professor position at École polytechnique, Centre de Mathématiques Appliquées in 2017 and co-chairs the Initiative HPC@Maths with a specific award from the Foundation of École polytechnique and which has become a team within the CMAP. He has also created the Computing Center at École polytechnique and has been awarded the E. Brun Award for the French Academia of Sciences in 2022. He is presently vice-chair of the applied mathematics department at École polytechnique. His main fields of research are mathematical modeling and numerical analysis, analysis of PDEs and dynamical systems for multi-scale systems, scientific computing and high-performance computing with applications in combustion, two-phase flows, plasma physics and biomedical engineering.
