1er Symposium canadien en analyse numérique et calcul scientifique
A Stabilized Meshfree, Kernel-Based Method for Convection-Diffusion Problems
Colin, Fabrice (1), Richard Egli (2), and Abdellatif Serghini Mounim (1)
(1) Département de mathématiques et d'informatique / Mathematics and Computer Science, Université Laurentienne / Laurentian University, (2) Département d’Informatique, Université de Sherbrooke

For many decades meshfree mehtods have been widely studied and developed since they were introduced for the first time in the field of numerical simulation methods. In this talk, we will present a meshfree, particle-based method for Convection-Diffusion problems. It is well known that standard approaches in the framework of finite differences, finite volumes, finite elements, and particle methods work well for elliptical differential equations. However, when applied without the use of a stabilization method to other classes of differential equations, they were found to yield unstable solutions, i.e. solutions that exhibit non-physical spatial oscillations. In our case, the discretization is performed by using piecewise constant kernels and the stabilized scheme we found, is based on a new upwind kernel. We will also show that accurate and stable schemes can be obtained by using special purpose-built kernels, based on a popular stabilization parameter from the finite difference method. In addition, we will establish that under some ideal conditions, the classical optimal finite difference scheme can be derived from the new method. Several numerical tests on four standard problems show the efficiency of the proposed method. Future developments and improvements will also be discussed.

Mercredi, 19 juin, 10h00
Salle Des Plaines C