Schedule

By hovering the cursor over a title, you can see the abstract of the corresponding presentation.

Thursday, January 22, 2009
 

08:45-09:15	Welcome
09:15-09:30	Introduction		M. Postel
09:30-10:15	Fully adaptive multiresolution methods for evolutionary PDEs We present efficient fully adaptive numerical schemes for evolutionary partial differential equations based on a finite volume (FV) discretization with explicit time discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. Embedded Runge-Kutta methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non-admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the finite volume scheme on a regular grid is reported, which demonstrates the efficiency of the new method.	(PDF)	K. Schneider
10:15-11:00	Multiresolution acceleration methods in three dimensions We show a version of Harten's cell-averaged multiresolution (MR) scheme that works on three-dimensional unstructured meshes. The MR method uses the difference in information between adjacent grid levels on a set of nested grids for determining active and inactive grid cells. This boils down to computing the wavelet decomposition known to be a rich source of regularity information. Shocks, contact discontinuities, reaction fronts, or any other inviscid and viscous flow features can be identified and tracked in a time-accurate fashion. In active regions the underlying finite volume scheme is solved in the usual manner, whereas in smooth regions an inexpensive interpolation of the numerical divergence replaces both flux computations and reconstruction. Thus the simulation becomes significantly more efficient without any loss of accuracy compared to the finest grid available. The combined finite volume-MR method will be described followed by 2- and 3-D examples on unstructured meshes.		B. L. Bihari
11:00-11:30	Coffee break
11:30-12:00	Parallelization of multiscale-based grid adaptation using space filling curves	(PDF)	S. Mogosan
12:00-12:30	Utilisation of Harten multiresolution in scientific computing: two examples	(PDF)	G. Chiavassa
12:30-14:00	Lunch
14:00-14:45	Combining multiresolution and anisotropy: theory, algorithms and open problems Multiresolution adaptive method are known to be of efficient use in the numerical solution of PDE's, providing with efficient compression and data structure. In these methods the local refinement of elements/volumes is usually isotropic. In this talk, we shall address the issue of deriving multiresolution adaptive methods based on anisotropic triangulations. We discuss the approximation theory available for such triangulations. When the function to be approximated is known to us, we propose a greedy algorithm which has the ability to generate adaptive hierarchical triangulations that exhibit a locally optimal aspect ratio, resulting in an optimal convergence rate. We finally discuss the difficulties which are to be dealt with when applying these ideas to PDE's.	(PDF)	A. Cohen
14:45-15:15	Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems We propose and study a posteriori error estimates for convection-diffusion-reaction problems approximated by weighted interior-penalty discontinuous Galerkin methods. Our estimates are fully robust in the singularly perturbed regimes for the error measured in a norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. This means that the effectivity index (overestimation factor) is independent of the size of the convection or reaction terms. The estimates do not involve any undetermined constants, are fully computable, and can thus be used to the actual error control. We achieve this feature by introducing H(div)-conforming diffusive and convective flux reconstructions. Finally, our estimates are locally efficient, whence they are also suitable for adaptive mesh refinement. Numerical experiments illustrate these theoretical results.	(PDF)	M. Vohralik
15:15-15:45	Coffee break
15:45-16:15	Simulation of incompressible flows with artificial compressibility using fully adaptive multiscale schemes		Y. Stiriba
16:15-16:45	A new strategy for adapting time-step in the Local Time Stepping method applied to hyperbolic PDEs We are concerned with the numerical simulations of two-phase flows representing oil transportation along a 1-D pipeline, over a long time-range. Such problems are modeled by a highly nonlinear system of partial differential equations of conservations laws. For this kind of evolutionary problems, we make use of the Lagrange-Projection finite-volume scheme coupled with the Multiresolution and the Local Time Stepping methods. Indeed, employing these two adaptive methods allows us to speed up the computation while maintaining the accuracy of the results. In this talk, we want to emphasize on the new strategy for computing the time-step in the context of the Local Time Stepping method applied to hyperbolic PDEs system. More specifically, instead of using constant "micro" time-steps, we propose to recalculate them after each local evolution of the solution on the adaptive grid. These varying adequate time-steps must decrease during each "macro" time-step in order to satisfy the stability of the numerical scheme. Numerical results as well as CPU time of realistic simulations will be presented to show the efficiency of such a strategy.	(PDF)	Q. L. Nguyen

Friday, January 23, 2009
 

09:30-10:15	Adaptive methods for the Vlasov equation In this talk we shall give a review of the work we have performed on the development of adaptive solvers for the Vlasov- Poisson and Vlasov-Maxwell equations. Different multiresolution techniques have been compared, in particular interpolating wavelets and hierarchical finite elements. We shall address the refinement strategies and also the data structures and efficient parallel implementation techniques which remain one of the bottlenecks of such solvers in higher dimensions. Our solutions in four dimensional phase space shall be detailed.	(PDF)	E. Sonnendrücker
10:15-11:00	Wavelet-based CVS method to solve a convection-dominated problem: the numerical simulation of turbulence	(PDF)	M. Farge
11:00-11:30	Coffee break
11:30-12:00	How to predict accurate grids in adaptive semi-Lagrangian schemes In this talk I will present a new class of adaptive semi-Lagrangian schemes -- based on performing a semi-Lagrangian method on adaptive interpolation grids -- in the context of solving nonlinear transport problems with underlying smooth characteristic flow. I will describe two frameworks for implementing adaptive interpolations, namely multilevel finite elements and interpolatory wavelets. For both discretizations, I introduce a notion of good adaptivity to a given function, and show that it is preserved by a low-cost prediction algorithm which transports multilevel grids along any smooth numerical flow. Error estimates are then established for the resulting ``predict and readapt'' schemes. As for the wavelet case, these results are new and also apply to high-order interpolation.	(PDF)	M. Campos-Pinto
12:00-12:30	Local time steps for a finite volume scheme We present a strategy for solving time-dependent problems on grids with local refinements in time using different time steps in different regions of space. We discuss and analyze two conservative approximations for a parabolic equation, based on finite volume with piecewise constant projections and domain decomposition techniques. Next we present an iterative method for solving the composite-grid system that reduces to solution of standard problems with standard time stepping on the coarse and fine grids. At every step of the algorithm, conservativity is ensured. Finally, numerical results illustrate the accuracy of the proposed methods.	(PDF)	I. Faille
12:30-14:00	Lunch
14:00-14:45	Using Harten’s multiresolution Framework on existing codes for hyperbolic PDEs	(PDF)	R. Donat
14:45-15:00	Closing remarks	S. Müller

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