Erwan Deriaz

Erwan Deriaz

This paper presents original and close to optimal stability conditions linking the time step and the space step, stronger than the CFL criterion: $δt\leq Cδx^α$ with $α=\frac{2r}{2r-1}$, $r$ an integer, for some numerical schemes we produce, when solving convection-dominated problems. We test this condition numerically and prove that it applies to nonlinear equations under smoothness assumptions.

Erwan Deriaz

We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we demonstrate that some explicit schemes are numerically stable for small perturbations under the condition $δt\leq C δx^{2r/(2r-1)}$ where $r$ is an integer, $δt$ the time step and $δx$ the space step.

Erwan Deriaz

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.