Space-time domain decomposition for advection-diffusion problems in mixed formulations

This paper is concerned with the numerical solution of porous-media flow and transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time steps can be used in different parts of the domain. Global-in-time, non-overlapping domain-decomposition methods are coupled with operator splitting making possible the different treatment of the advection and diffusion terms. Two domain-decomposition methods are considered: one uses the time-dependent Steklov--Poincar{\'e} operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains. A generalized Neumann-Neumann preconditioner is proposed for the first method. To illustrate the two methods numerical results for two-dimensional problems with strong heterogeneities are presented. These include both academic problems and more realistic prototypes for simulations for the underground storage of nuclear waste

A Schwarz Waveform Relaxation Method for Advection—Diffusion—Reaction Problems with Discontinuous Coefficients and Non-matching Grids

International audienceWe present a non-overlapping Schwarz waveform relaxation method for solving advection-reaction-diffusion problems in heterogeneous media. The do-main decomposition method is global in time, which permits the use of different time steps in different subdomains. We determine optimal non-local, and optimized Robin transmission conditions. We also present a space-time finite volume scheme es-pecially designed to handle such transmission conditions. We show the performance of the method on an example inspired from nuclear waste disposal simulations

Three-dimensional paraxial migration method without lateral splitting

International audienceWe introduce a migration algorithm based on paraxial wave equation that does not use any splitting in the lateral variables. The discretization is first derived in the constant coefficient case by higher order finite differences, then generalized to arbitrarily varying velocities via finite elements. We present a detailed plane wave analysis in a homogeneous medium, and give evidence that numerical dispersion and anisotropy can be controlled. Propagation along depth is done with a higher order method based on a conservative Runge Kutta method. At each step in depth we have to solve a large linear system. This is the most time consuming part of the method. The key to obtaining good performance lies in the use of a Conjugate Gradient like iterative solver. We show the performance of the method with a model example

Solution to the SIAM «Hundred-dollar, Hundred-digit Challenge»

In February 2002, L. N. Trefethen proposed a lis of 10 short problems, each with a numerical answer. The challenge was to compute each of the numbers to 10 digits accuracy. This report gives a solution to each of the 10 problems. The problems range to the computation of an integral, to optimizing oscillatory functions, through partial differnetial equations and probability theory. We detail the methods used in each case, and comment on how we obtained the requested accuracy

Parallel Solution of the Wave Equation Using Higher Order Finite Elements

International audienceThe time domain simulation of wave propagation phenomena is a computationally demanding task. The acoustic wave equation is the simplest such model and serves as a useful benchmark for more realistic situations (elastodynamics, or electromagnetism). This paper presents a parallel simulation code for such phenomena. The initial implementation is for 2D acoustics, but of course the method is general, and we are currently investigating more complex models.We use the higher order finite elements developed by Cohen et al.. These elements were designed to give a diagonal mass matrix, thus enabling an explicit solution, while retaining high accuracy. They are based on a modification of the classical degree 2 and 3 elements. We also recall how the modified equation technique leads to higher order methods in time. As the resulting method is explicit, it lends itself very naturally to a parallel implementation. We have chosen a coarse grain, domain splitting approach, using message passing, as this is known to be the most portable approach, likely to give the best efficiency on a wide range of parallel computers

The COUPLEX Test Cases: Nuclear Waste Disposal Simulation

International audienceThe models appearing in the COUPLEX benchmark are a set of simplified albeit realistic test cases aimed at simulating the transport of radionuclides around a nuclear waste repository. Three different models were used: The first test case is related to simulations based on a simplified 2D far-field model close to those used for safety assessments in nuclear waste management. It leads to a classical convec-tion diffusion type problem, but with highly variable parameters in space, highly concentrated sources in space and time, very different time scales and accurate results expected even after millions of years. The second test case is a simplification of a typical 3D near-field computation, taking into account the glass dissolution of vitrified waste, and the congruent release of several radionu-clides (including daughter products), with their migration through the geological barrier. The aim of the third test case is to use the results of the near-field computation (COUPLEX 2) to drive the behavior of the nuclide source term in the Far Field computation (COUPLEX 1). The modeling of this last case was purposely left rather open, unlike the previous two, leaving the choice to participants of the way the coupling should be made

Space-time Domain Decomposition and Mixed Formulation for solving reduced fracture models

International audienceIn this paper we are interested in the "fast path" fracture and we aim to use global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium containing such a fracture. We consider a reduced model in which the fracture is treated as an interface between the two subdomains. Two domain decomposition methods are considered: one uses the time-dependent SteklovPoincaré operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Ventcell transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains and in the fracture. Demonstrations of the well-posedness of the Ventcell subdomain problems is given for the mixed formulation. An analysis for the convergence factor of the OSWR algorithm is given in the case with fractures to compute the optimized parameters. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods

Block Preconditioning for a Coupled Model of Transport with Sorption in Porous Media

Session 5International audienc

Imagerie du proche sous-sol par un radar géologique

Ce travail a pour objet la modélisation numérique de la propagation des ondes électromagnétiques dans le proche sous-sol, à des fins de repérage d'objets enfouis, et de produire des images du sous-sol, en utilisant des méthodes inspirées de la sismique. La modélisation utilise la méthode de Yee pour résoudre les équations de Maxwell 2D par différences finies dans le domaine temporel. Grâce aux techniques d'états adjoints, on peut ensuite calculer des images du sous-sol, montrant où se trouvent des objets enterrés. Le caractère conducteur du sous-sol rendant le phénomène irréversible en temps, on utilise un technique due à Griewank pour minimiser le nombre de simulations, compte tenu de la place mémoire disponible

Parallel solution of the wave equation using higher order finite elements

International audienceWe present a parallel solver for wave propagation problems based on the higher-order explicit finite elements developed by Cohen et al. (1995) These elements were introduce to allow mass-lumping while preserving high accuracy. Our approach is based on a coarse-grain, domain-splitting parallelism, and uses the new MPI (Message Passing Interface) standard as a message passing library. The program currently runs on a network of workstations, on a Cray T3D and on an IBM SP/