Sébastien Boyaval

Sébastien Boyaval, Claude Le Bris, Tony Lelièvre et al.

We report here on the recent application of a now classical general reduction technique, the Reduced-Basis approach initiated in [C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A. T. Patera, and G. Turinici. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. Journal of Fluids Engineering, 124(1):7080, 2002.], to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: [S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen, and A.T. Patera. A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin co-efficient. Computer Methods in Applied Mechanics and Engineering, 198(4144):3187-3206, 2009.], which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and [S. Boyaval and T. Lelièvre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm with T. Lelièvre, Commun. Math. Sci. 8, special Issue "Mathematical Issue on Complex Fluids" P. Zhang ed., to appear, 2010, ARXIV preprint arXiv:0906.3600], which uses a RB type approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction.

Sébastien Boyaval, Tony Lelièvre, Claude Mangoubi

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.

John Barrett, Sébastien Boyaval

We 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 $D $\subset$ R d , d = 2 or 3$, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics ($d = 2$) or a reduced version, where the tangential component on each simplicial edge ($d = 2$) or face ($d = 3$) 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 $d = 2$, 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.

Sébastien Boyaval

The Reduced-Basis Control-Variate Monte-Carlo method was introduced recently in [S. Boyaval and T. Lelièvre, 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.

François Bouchut, Sébastien Boyaval

We 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.

Sébastien Boyaval

We pursue here the development of models for complex (viscoelastic) fluids in shallow free-surface gravity flows which was initiated by [Bouchut-Boyaval, M3AS (23) 2013] for 1D (translation invariant) cases. The models we propose are hyperbolic quasilinear systems that generalize Saint-Venant shallow-water equations to incompressible Maxwell fluids. The models are compatible with a formulation of the thermo-dynamics second principle. In comparison with Saint-Venant standard shallow-water model, the momentum balance includes extra-stresses associated with an elastic potential energy in addition to a hydrostatic pressure. The extra-stresses are determined by an additional tensor variable solution to a differential equation with various possible time rates. For the numerical evaluation of solutions to Cauchy problems, we also propose explicit schemes discretizing our generalized Saint-Venant systems with Finite-Volume approximations that are entropy-consistent (under a CFL constraint) in addition to satisfy exact (discrete) mass and momentum conservation laws. In comparison with most standard viscoelastic numerical models, our discrete models can be used for any retardation-time values (i.e. in the vanishing "solvent-viscosity" limit). We finally illustrate our hyperbolic viscoelastic flow models numerically using computer simulations in benchmark test cases. On extending to Maxwell fluids some free-shear flow testcases that are standard benchmarks for Newtonian fluids, we first show that our (numerical) models reproduce well the viscoelastic physics, phenomenologically at least, with zero retardation-time. Moreover, with a view to quantitative evaluations, numerical results in the lid-driven cavity testcase show that, in fact, our models can be compared with standard viscoelastic flow models in sheared-flow benchmarks on adequately choosing the physical parameters of our models. Analyzing our models asymptotics should therefore shed new light on the famous High-Weissenberg Number Problem (HWNP), which is a limit for all the existing viscoelastic numerical models.

Sébastien Boyaval

The shallow-water equations of Saint-Venant, often used to model the long-wave dynamics of free-surface flows driven by inertia and hydrostatic pressure, can be generalized to account for the elongational rheology of non-Newtonian fluids too. We consider here the $4 \times 4$ shallow-water equations generalized to viscoelastic fluids using the Johnson-Segalman model in the elastic limit (i.e. at infinitely-large Deborah number, when source terms vanish). The system of nonlinear first-order equations is hyperbolic when the slip parameter is small $ζ\le 1/2$ ($ζ$ = 1 is the corotational case and $ζ= 0$ the upper-convected Maxwell case). Moreover, it is naturally endowed with a mathematical entropy (a physical free-energy). When $ζ\le 1/2$ and for any initial data excluding vacuum, we construct here, when elasticity $G > 0$ is non-zero, the unique solution to the Riemann problem under Lax admissibility conditions. The standard Saint-Venant case is recovered when $G \to 0$ for small data.

Sébastien Boyaval

Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in [Bouchut \& Boyaval, 2013], which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solution to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but numerical simulations went smoothly in a practically useful range of parameters.

Jean-Paul Travert, Cédric Goeury, Sébastien Boyaval et al.

Flood mapping and water depth estimation from Synthetic Aperture Radar (SAR) imagery are crucial for calibrating and validating hydraulic models. This study uses SAR imagery to evaluate various preprocessing (especially speckle noise reduction), flood mapping, and water depth estimation methods. The impact of the choice of method at different steps and its hyperparameters is studied by considering an ensemble of preprocessed images, flood maps, and water depth fields. The evaluation is conducted for two flood events on the Garonne River (France) in 2019 and 2021, using hydrodynamic simulations and in-situ observations as reference data. Results show that the choice of speckle filter alters flood extent estimations with variations of several square kilometers. Furthermore, the selection and tuning of flood mapping methods also affect performance. While supervised methods outperformed unsupervised ones, tuned unsupervised approaches (such as local thresholding or change detection) can achieve comparable results. The compounded uncertainty from preprocessing and flood mapping steps also introduces high variability in the water depth field estimates. This study highlights the importance of considering the entire processing pipeline, encompassing preprocessing, flood mapping, and water depth estimation methods and their associated hyperparameters. Rather than relying on a single configuration, adopting an ensemble approach and accounting for methodological uncertainty should be privileged. For flood mapping, the method choice has the most influence. For water depth estimation, the most influential processing step was the flood map input resulting from the flood mapping step and the hyperparameters of the methods.

Sébastien Boyaval

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.