516 research outputs found
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
A fast Monte-Carlo method with a Reduced Basis of Control Variates applied to Uncertainty Propagation and Bayesian Estimation
The Reduced-Basis Control-Variate Monte-Carlo method was introduced recently
in [S. Boyaval and T. Leli\`evre, CMS, 8 2010] as an improved Monte-Carlo
method, for the fast estimation of many parametrized expected values at many
parameter values. We provide here a more complete analysis of the method
including precise error estimates and convergence results. We also numerically
demonstrate that it can be useful to some parametrized frameworks in
Uncertainty Quantification, in particular (i) the case where the parametrized
expectation is a scalar output of the solution to a Partial Differential
Equation (PDE) with stochastic coefficients (an Uncertainty Propagation
problem), and (ii) the case where the parametrized expectation is the Bayesian
estimator of a scalar output in a similar PDE context. Moreover, in each case,
a PDE has to be solved many times for many values of its coefficients. This is
costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C.
Le Bris, Nguyen C., Y. Maday and T. Patera, CMAME, 198 2009]. This is the first
combination of various Reduced-Basis ideas to our knowledge, here with a view
to reducing as much as possible the computational cost of a simple approach to
Uncertainty Quantification
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
Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems
This work focuses on providing accurate low-cost approximations of stochastic Âżnite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments.Postprint (author's final draft
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
Finite element approximation of the FENE-P model
International audienceWe extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain , subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics () or a reduced version, where the tangential component on each simplicial edge () or face () is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes, based on the backward Euler type time discretiza-tion, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case , as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions
A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Basis Paradigm
In this work, we develop a reduced-basis approach for the efficient
computation of parametrized expected values, for a large number of parameter
values, using the control variate method to reduce the variance. Two algorithms
are proposed to compute online, through a cheap reduced-basis approximation,
the control variates for the computation of a large number of expectations of a
functional of a parametrized Ito stochastic process (solution to a parametrized
stochastic differential equation). For each algorithm, a reduced basis of
control variates is pre-computed offline, following a so-called greedy
procedure, which minimizes the variance among a trial sample of the output
parametrized expectations. Numerical results in situations relevant to
practical applications (calibration of volatility in option pricing, and
parameter-driven evolution of a vector field following a Langevin equation from
kinetic theory) illustrate the efficiency of the method
- …