A posteriori analysis of Chorin-Temam scheme for Stokes equations

We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time-discretization of an unstationary Stokes problem. Inspired by the analyses of the Backward Euler scheme performed by C.Bernardi and R.Verf\"urth, we derive a posteriori estimators for the error on the velocity gradient in L2 norm. Our invesigation is supported by numerical experiments

A new model for shallow viscoelastic free-surface flows forced by gravity on rough inclined bottom

International audienceA thin-layer model for shallow viscoelastic free-surface gravity flows on slippery topogra-phies around a flat plane has been derived recently in [Bouchut-Boyaval, M3AS (23) 2013]. We show here how the model can be modified for flows on rugous topographies varying around an inclined plane. The new reduced model extends the scope of one derived in [Bouchut-Boyaval, M3AS (23) 2013]. It is one particular thin-layer model for free-surface gravity flows among many ones that can be formally derived with a generic unifying procedure. Many rheologies and various shallow flow regimes have already been treated within a single unified framework in [Bouchut-Boyaval, HAL-ENPC (00833468) 2013]. The initial full model used here as a starting point is however a little different to one used in [Bouchut-Boyaval, HAL-ENPC (00833468) 2013], although the new thin-layer model is very similar to the one derived therein. Precisely, here, the bulk dissipation (due to e.g. viscosity) is neglected from the beginning, like in [Bouchut-Boyaval, M3AS (23) 2013]. Moreover, unlike in [Bouchut-Boyaval, HAL-ENPC (00833468) 2013], we perform here numerical simulations. The interest of the extension is illustrated in a physically interesting situation where new stationary solutions exist. To that aim, the Finite-Volume method proposed in [Bouchut-Boyaval, M3AS (23) 2013] needs to be modified, with an adequate discretization of the new source terms. Interestingly, we can also numerically exhibit an apparently new kind of "roll-wave" solution

Free-energy-dissipative schemes for the Oldroyd-B model

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman, which have been reported to be numerically more stable than discretizations of the usual formulation in some benchmark problems. Our analysis gives some tracks to understand these numerical observations

A new model for shallow viscoelastic fluids

International audienceWe propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver

A new framework for convergence studies in TELEMAC-2D

Water Qualit

Reduced-Basis approach for homogenization beyond the periodic setting

We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the macroscopic scale is required at the microscopic scale. This is a framework very much adapted to model order reduction attempts. The purpose of this work is to show how the reduced-basis approach allows to speed up the computation of a large number of cell problems without any loss of precision. The essential components of this reduced-basis approach are the {\it a posteriori} error estimation, which provides sharp error bounds for the outputs of interest, and an approximation process divided into offline and online stages, which decouples the generation of the approximation space and its use for Galerkin projections

On the Modeling and Simulation of Non-Hydrostatic Dam Break Flows

International audienceThe numerical simulation of three-dimensional dam break flows is discussed. A non-hydrostatic numerical model for free-surface flows is considered, which is based on the incompressible Navier-Stokes equations coupled with a volume-of-fluid approach. The numerical results obtained for a variety of benchmark problems show the validity of the numerical approach, in comparison with other numerical models, and allow to investigate numerically the non-hydrostatic three-dimensional effects, in particular for the usual test cases where hydrostatic approximations are known analytically. The numerical experiments on actual topographies, in particular the Malpasset dam break and the (hypothetical) break of the Grande-Dixence dam in Switzerland, also illustrate the capabilities of the method for large-scale simulations and real-life visualization

Approximation Volumes-Finis de la mesure invariante d'une loi de conservation scalaire stochastique visqueuse

We aim to give a numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable, and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz and to have at most polynomial growth. The numerical scheme we employ discretises the SPDE according to a finite volume method in space, and a split-step backward Euler method in time. As a first result, we prove the well-posedness as well as the existence and uniqueness of an invariant measure for both the spatial semi-discretisation and the fully discrete scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. A few numerical experiments are performed to illustrate these results