Sébastien BOYAVAL

Directeur LHSV

Ecole des Ponts ParisTech / Laboratoire d'Hydraulique Saint-Venant

Member of Team project MATHERIALS Inria Paris

Fields of interest

Numerical analysis ; Partial Differential Equations ; Continuum Mechanics ; Rheology

Theses supervised

Arthur Guillot-LeGoff (2021-...) with Brigitte Vincon-Leite (ENPC)
Jean-Paul Travert (2021-...)
Elisa Béteille (2021-...) with Frédérique Larrarte (UGE)
Jana Tarhini (2021-...) with Quang Huy Tran (IFPEN)
Sofiane Martel (2016-2019)
Riad Sanchez (2014-2017) with Quang Huy Tran (IFPEN)

Publications

Construction and performance of kinetic schemes for linear systems of conservation laws
- Audusse Emmanuel
- Boyaval Sébastien
- Dubos Virgile
- Le Minh-Hoang
, 2025 . We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the construction of kinetic schemes that satisfy a discrete equivalent to a convex extension (an additional non-trivial conservation law) of the target system -the (linear) acoustic and elastodynamics systems, specifically -. Then, we evaluate numerically the convergence of various possible kinetic schemes toward smooth solutions, in comparison with standard finite-difference and finite-volume discretizations on Cartesian meshes. Our numerical results confirm the interest of ensuring a discrete equivalent to a convex extension, and show the influence of remaining parameter variations in terms of error magnitude, both for "first-order" and "second-order" kinetic schemes : the parameter choice with largest CFL number (equiv., smallest spurious diffusion in the equivalent equation analysis) has the smallest discretization error.
Global solutions and uniform convergence stability for compressible Navier-Stokes equations with oldroyd-type constitutive law
- Wang Na
- Boyaval Sébastien
- Hu Yuxi
Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag , 2025 . We consider one dimensional isentropic compressible Navier-Stokes equations with Oldroyd-type constitutive law. By establishing uniform a priori estimates (with respect to relaxation time), we show global existence of smooth solutions with small initial data. Moreover, we get global-in-time convergence of the system towards the classical isentropic compressible Navier-Stokes equations. (10.1002/mana.70075)
DOI : 10.1002/mana.70075
Blowup of solutions for compressible viscoelastic fluid
- Wang Na
- Boyaval Sébastien
- Hu Yuxi
Procedia Materials Science (Elsevier), Elsevier , 2025 . We prove finite-time blowup of classical solutions for the compressible Upper Convective Maxwell (UCM) viscoelastic fluid system. By establishing a key energy identity and adapting Sideris' method for compressible flows, we derive a Riccati-type inequality for a momentum functional. For initial data with compactly supported perturbations satisfying a sufficiently large condition, all classical solutions lose regularity in finite time. This constitutes the first rigorous blowup result for multidimensional compressible viscoelastic fluids. (10.1016/j.aml.2025.109774)
DOI : 10.1016/j.aml.2025.109774
Dam-break flow over various obstacles configurations
- Beteille Elisa
- Larrarte Frederique
- Boyaval Sebastien
- Demay Eric
- Le Minh-Hoang
Journal of Ecohydraulics, Taylor & Francis , 2025, 63, pp.156-170 . Fast floods resulting from the failure of hydraulic structures can be characterized by ‘dam-break’ type waves. They pose catastrophic risks to downstream populations and result in severe structural damage, especially in urban areas. To assess and mitigate these risks, it is essential to forecast the influence of urban forms on flooding severity at a global scale. This paper provides datasets from reduced-scale physical experiments of transient flow through various obstacles configurations. The experiments are conducted in a rectangular horizontal open channel, where flow conditions are achieved by rapidly opening a gate holding a volume of water. To assess the impact of obstacle configurations on flow behaviour, two obstacle sizes are investigated, along with one idealized city layout. The experiments provide complete water hydrographs upstream and downstream of the gate. Additionally, the good performance of the code_saturne computational fluid dynamics (CFD) solver and the volume-of-fluid (VOF) method in numerically simulating the experiments is demonstrated. (10.1080/00221686.2025.2460020)
DOI : 10.1080/00221686.2025.2460020
Experimental analysis on the influence of urban forms on unsteady urban flooding
- Beteille Elisa
- Boyaval Sébastien
- Larrarte Frédérique
- Demay Eric
, 2025 . Unsteady urban flooding, such as dam-break waves, poses catastrophic risks to downstream populations and results in severe damage. To assess and mitigate these risks, it is essential to forecast the influence of urban forms on flooding severity. In this paper, datasets are provided from reduced-scale physical experiments of transient flow through idealized suburban districts. The experiments are conducted in a rectangular horizontal open channel, where flow conditions are achieved by rapidly opening a gate holding a volume of water. To evaluate the impact of urban forms on flow behavior, we investigated two urban parameters; the number and width of streets in the main direction of the flow. The experiments provide complete water hydrographs for thirteen urban forms. Conductive and acoustic gauges are positioned at different locations to track the wavefront and water depth variation. The results illustrate the impact of the two studied urban parameters on flow variables and provide valuable validation data for computational urban planning models.
About the structural stability of Maxwell fluids: convergence toward elastodynamics
- Boyaval Sébastien
, 2022 . Maxwell's models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But rigorous proofs are lacking. The present note is a contribution toward well-defined viscoelastic flows proved to encompass both solid and (liquid) fluid regimes. In a first part, we consider the structural stability of particular viscoelastic flows: 1D shear waves solutions to damped wave equations. We show the convergence toward purely elastic 1D shear waves solutions to standard wave equations, as the relaxation time λ and the viscosity µ grow unboundedly λ ≡ µ/G → ∞ in Maxwell's constitutive equation λ τ +τ = 2 µD(u) for the stress τ of viscoelastic fluids with velocity u. In a second part, we consider the structural stability of general multi-dimensional viscoelastic flows. To that aim, we embed Maxwell’s constitutive equation in a symmetric-hyperbolic system of PDEs which we proposed in our previous publication [ESAIM:M2AN 55 (2021) 807-831] so as to define multi-dimensional viscoelastic flows unequivocally. Next, we show the continuous dependence of multi-dimensional viscoelastic flows on λ ≡ μ/ G using the relative-entropy tool developped for symmetric-hyperbolic systems after C. M. Dafermos. It implies convergence of the viscoelastic flows defined in [ESAIM:M2AN 55 (2021) 807-831] toward compressible neo-Hookean elastodynamics when λ → ∞.
A viscoelastic flow model of maxwell-type with a symmetric-hyperbolic formulation
- Boyaval Sébastien
, 2022 . Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021) 807-831] an upper-convected Maxwell model for compressible flows with a symmetrichyperbolic formulation. Here, that model is derived again, with new details.
Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law
- Boyaval Sébastien
- Martel Sofiane
- Reygner Julien
IMA Journal of Numerical Analysis, Oxford University Press (OUP) , 2022, 42 (3), pp.2710-2770 . We study the 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 continuous 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 semi-discrete and the split-step 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. We investigate rates of convergence theoretically, in the case where the flux function is globally Lipschitz continuous with a small Lipschitz constant, and numerically for the Burgers equation. (10.1093/imanum/drab049)
DOI : 10.1093/imanum/drab049
Viscoelastic flows of Maxwell fluids with conservation laws
- Boyaval Sébastien
ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP , 2021, 55 (3), pp.807-831 . We consider multi-dimensional extensions of Maxwell's seminal rheo-logical equation for 1D viscoelastic flows. We aim at a causal model for compressible flows, defined by semi-group solutions given initial conditions , and such that perturbations propagates at finite speed. We propose a symmetric hyperbolic system of conservation laws that contains the Upper-Convected Maxwell (UCM) equation as causal model. The system is an extension of polyconvex elastodynamics, with an additional material metric variable that relaxes to model viscous effects. Interestingly, the framework could also cover other rheological equations, depending on the chosen relaxation limit for the material metric variable. We propose to apply the new system to incompressible free-surface gravity flows in the shallow-water regime, when causality is important. The system reduces to a viscoelastic extension of Saint-Venant 2D shallow-water system that is symmetric-hyperbolic and that encompasses our previous viscoelastic extensions of Saint-Venant proposed with F. Bouchut. (10.1051/m2an/2020076)
DOI : 10.1051/m2an/2020076
Non-isothermal viscoelastic flows with conservation laws and relaxation
- Boyaval Sébastien
- Dostalík Mark
, 2022, pp.337-364 . We propose a system of conservation laws with relaxation source terms (i.e. balance laws) for non-isothermal viscoelastic flows of Maxwell fluids. The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using additional structure variables. It is obtained by writing the Helmholtz free energy as the sum of a volumetric energy density (function of the determinant of the deformation gradient det F and the temperature θ like the standard perfect-gas law or Noble-Abel stiffened-gas law) plus a polyconvex strain energy density function of F, θ and of symmetric positive-definite structure tensors that relax at a characteristic time scale. One feature of our model is that it unifies various ideal materials ranging from hyperelastic solids to perfect fluids, encompassing fluids with memory like Maxwell fluids. We establish a strictly convex mathematical entropy to show that the system is symmetric-hyperbolic. Another feature of the proposed model is therefore the short-time existence and uniqueness of smooth solutions, which define genuinely causal viscoelastic flows with waves propagating at finite speed. In heat-conductors, we complement the system by a Maxwell-Cattaneo equation for an energy-flux variable. The system is still symmetric-hyperbolic, and smooth evolutions with finite-speed waves remain well-defined. (10.1142/S0219891622500096)
DOI : 10.1142/S0219891622500096
Viscoelastic flows with conservation laws
- Boyaval Sébastien
, 2019 . We propose in this work the first symmetric hyperbolic system of conservation laws to describe viscoelastic flows of Maxwell fluids, i.e. fluids with memory that are characterized by one relaxation-time parameter. Precisely, the system of quasilinear PDEs is detailed for the shallow- water regime, i.e. for hydrostatic incompressible 2D flows with free surface under gravity. It generalizes Saint-Venant system to viscoelastic flows of Maxwell fluids, and encompasses previous works with F. Bouchut. It also generalizes the (thin-layer) elastodynamics of hyperelastic materials to viscous fluids, and to various rheologies between solid and liquid states that can be formulated using our new variable as material parameter. The new viscoelastic flow model has many potential applications, additionally to falling into the theoretical framework of (symmetric hyper- bolic) systems of conservation laws. In computational rheology, it offers a new approach to the High-Weissenberg Number Problem (HWNP). For transient geophysical flows, it offers perspectives of thermodynamically-compatible numerical simulations, with a Finite-Volume (FV) discretization say. Besides, one FV discretization of the new continuum model is proposed herein to precise our ideas incl. the physical meaning of the solutions. Perspectives are finally listed after some numerical simulations.
A semi-Lagrangian splitting method for the numerical simulation of sediment transport with free surface flows
- Boyaval Sébastien
- Caboussat Alexandre
- Mrad Arwa
- Picasso Marco
- Steiner Gilles
Computers and Fluids, Elsevier , 2018, 172, pp.384-396 . We present a numerical model for the simulation of 3D poly-dispersed sediment transport in a Newtonian flow with free surfaces. The physical model is based on a mixture model for multiphase flows. The Navier–Stokes equations are coupled with the transport and deposition of the particle concentrations, and a volume-of-fluid approach to track the free surface between water and air. The numerical algorithm relies on operator-splitting to decouple advection and diffusion phenomena. Two grids are used, based on unstructured finite elements for diffusion and an appropriate combination of the characteristics method with Godunov’s method for advection on a structured grid. The numerical model is validated through numerical experiments. Simulation results are compared with experimental results in various situations for mono-disperse and bi-disperse sediments, and the calibration of the model is performed using, in particular, erosion experiments. (10.1016/j.compfluid.2018.04.002)
DOI : 10.1016/j.compfluid.2018.04.002
Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation
- Kaercher Mark
- Boyaval Sébastien
- Grepl Martin
- Veroy Karen
Optimization and Engineering, Springer Verlag , 2018, 2018 (3) . We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach. (10.1007/s11081-018-9389-2)
DOI : 10.1007/s11081-018-9389-2
Derivation and numerical approximation of hyperbolic viscoelastic flow systems: Saint-Venant 2D equations for Maxwell fluids
- Boyaval Sébastien
, 2017 . 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.
Finite element approximation of the FENE-P model
- Barrett John W
- Boyaval Sébastien
IMA Journal of Numerical Analysis, Oxford University Press (OUP) , 2017 . 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 $D ⊂ R d , d = 2 or 3$, 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. (10.1093/imanum/drx061)
DOI : 10.1093/imanum/drx061
A Reduced-Basis Approach to Two-Phase Flow in Porous Media
- Tran Quang Huy
- Enchéry Guillaume
- Sanchez Riad
- Boyaval Sébastien
, 2017 . Reduced-basis methods (RB) have demonstrated their efficiency for a wide variety of problems, most of which are elliptic PDEs solved by finite element methods. In this work, we attempt to apply the RB philosophy to a simple “real-life” model for two-phase flows in porous media, whose reference scheme is a finite volume method. This model is parameterized by the viscosity of water. Because of the mixed parabolic-elliptic nature of the system, we first propose to restrict the RB approach to the pressure subsystem corresponding to the end time. The resulting parametric dependence is, however, much more intricate than in the classical examples. This difficulty will be discussed and illustrated by numerical results. (10.1007/978-3-319-57394-6_50)
DOI : 10.1007/978-3-319-57394-6_50
Polynomial Surrogates for Open-Channel Flows in Random Steady State
- El Moçayd Nabil
- Ricci Sophie
- Goutal Nicole
- Rochoux Mélanie C.
- Boyaval Sébastien
- Goeury Cédric
- Lucor Didier
- Thual Olivier
Environmental Modeling & Assessment, Springer , 2017, 23, pp.309–331 . Assessing epistemic uncertainties is considered as a milestone for improving numerical predictions of a dynamical system. In hydrodynamics, uncertainties in input parameters translate into uncertainties in simulated water levels through the shallow water equations. We investigate the ability of generalized polynomial chaos (gPC) surrogate to evaluate the probabilistic features of water level simulated by a 1-D hydraulic model (MASCARET) with the same accuracy as a classical Monte Carlo method but at a reduced computational cost. This study highlights that the water level probability density function and covariance matrix are better estimated with the polynomial surrogate model than with a Monte Carlo approach on the forward model given a limited budget of MASCARET evaluations. The gPC-surrogate performance is first assessed on an idealized channel with uniform geometry and then applied on the more realistic case of the Garonne River (France) for which a global sensitivity analysis using sparse least-angle regression was performed to reduce the size of the stochastic problem. For both cases, Galerkin projection approximation coupled to Gaussian quadrature that involves a limited number of forward model evaluations is compared with least-square regression for computing the coefficients when the surrogate is parameterized with respect to the local friction coefficient and the upstream discharge. The results showed that a gPC-surrogate with total polynomial degree equal to 6 requiring 49 forward model evaluations is sufficient to represent the water level distribution (in the sense of the ℓ2 norm), the probability density function and the water level covariance matrix for further use in the framework of data assimilation. In locations where the flow dynamics is more complex due to bathymetry, a higher polynomial degree is needed to retrieve the water level distribution. The use of a surrogate is thus a promising strategy for uncertainty quantification studies in open-channel flows and should be extended to unsteady flows. It also paves the way toward cost-effective ensemble-based data assimilation for flood forecasting and water resource management. (10.1007/s10666-017-9582-2)
DOI : 10.1007/s10666-017-9582-2
A Finite-Volume discretization of viscoelastic Saint-Venant equations for FENE-P fluids
- Boyaval Sébastien
, 2017, 200, pp.163-170 . 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. (10.1007/978-3-319-57394-6_18)
DOI : 10.1007/978-3-319-57394-6_18
An application of the Reduced-Basis Method for Darcy flows
- Sanchez Riad
- Boyaval Sébastien
- Enchéry Guillaume
- Tran Quang Huy
, 2016 .
Johnson-Segalman – Saint-Venant equations for viscoelastic shallow flows in the elastic limit
- Boyaval Sébastien
, 2017 . 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 × 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 ≤ 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 ≤ 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 → 0 for small data.
Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids
- Bouchut François
- Boyaval Sébastien
European Journal of Mechanics - B/Fluids, Elsevier , 2016, 55 (1), pp.116-131 . We propose a unified framework to derive thin-layer reduced models for some shallow free-surface flows driven by gravity. It applies to incompressible homogeneous fluids whose momentum evolves according to Navier-Stokes equations, with stress satisfying a rheology of viscous type (i.e. the standard Newtonian law with a constant viscosity, but also non-Newtonian laws generalized to purely viscous fluids and to viscoelastic fluids as well). For a given rheology, we derive various thin-layer reduced models for flows on a rugous topography slowly varying around an inclined plane. This is achieved thanks to a coherent simplification procedure, which is formal but based on a mathematically clear consistency requirement between scaling assumptions and the approximation errors in the differential equations. The various thin-layer reduced models are obtained depending on flow regime assumptions (either fast/inertial or slow/viscous). As far as we know, it is the first time that the various thin-layer reduced models investigated here are derived within the same mathematical framework. Furthermore, we obtain new reduced models in the case of viscoelastic non-Newtonian fluids, which extends [Bouchut & Boyaval, M3AS (23) 8, 2013]. (10.1016/j.euromechflu.2015.09.003)
DOI : 10.1016/j.euromechflu.2015.09.003
NUMERICAL SIMULATIONS OF THE PERIODIC INVISCID BURGERS EQUATION WITH STOCHASTIC FORCING
- Audusse Emmanuel
- Boyaval Sébastien
- Gao Yueyuan
- Hilhorst Danielle
, 2015, 48, pp.308-320 . We perform numerical simulations in the one-dimensional torus for the ﬁrst order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various egularities in space. For the numerical tests, we apply a ﬁnite volume scheme combining the Godunov numerical ﬂux with the Euler-Maruyama integrator in time. Our Monte-Carlo simulations are analyzed in bounded time intervals as well as in the large time limit, for various regularities in space. The empirical mean always converges to the space-average of the (deterministic) initial condition as t → ∞, just as the solution of the deterministic problem without source term, even if the stochastic source term is very rough. The empirical variance also stablizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise.
Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations
- Aghili Joubine
- Boyaval Sébastien
- Di Pietro Daniele
Computational Methods in Applied Mathematics, De Gruyter , 2015 . This paper presents two novel contributions on the recently introduced Mixed High-Order (MHO) methods [`Arbitrary order mixed methods for heterogeneous anisotropic diffusion on general meshes', preprint (2013)]. We first address the hybridization of the MHO method for a scalar diffusion problem and obtain the corresponding primal formulation. Based on the hybridized MHO method, we then design a novel, arbitrary order method for the Stokes problem on general meshes. A full convergence analysis is carried out showing that, when independent polynomials of degree k are used as unknowns (at elements for the pressure and at faces for each velocity component), the energy-norm of the velocity and the L2-norm of the pressure converge with order (k + 1), while the L2-norm of the velocity (super-)converges with order (k + 2). The latter property is not shared by other methods based on a similar choice of unknowns. The theoretical results are numerically validated in two space dimensions on both standard and polygonal meshes. (10.1515/cmam-2015-0004)
DOI : 10.1515/cmam-2015-0004
Numerical Simulation of the Dynamics of Sedimentary River Beds with a Stochastic Exner Equation
- Audusse Emmanuel
- Boyaval Sébastien
- Goutal Nicole
- Jodeau Magali
- Ung Philippe
ESAIM: Proceedings and Surveys, EDP Sciences , 2015, 48, pp.321 - 340 . At the scale of a river reach, the dynamics of the river bed is typically modelled by Exner equation (conservation of the solid mass) with an empirical solid flux of transported sediments, which is a simple deterministic algebraic formula function of i) the sediment physical characteristics (size and mass) and of ii) the averaged hydrodynamical description of the ambient water flow. This model has proved useful, in particular through numerical simulations, for hydraulic engineering purposes (like estimating the mass of sediments that is drained through an open dam). Though, the model is also coarse. And its applicability at various space and time scales remains a question of considerable interest for sedimentologists. In particular, physical experiments from the grain scale to the laboratory scale reveal important fluctuations of the solid flux in given hydrodynamical conditions. This work is a preliminary study of the coupling of a stochastic Exner equation with a hydrody-namical model for large scales. (Stochastic models with a probabilistic solid flux are currently being investigated, but most often only from the viewpoint of theoretical physics at the grain scale.) We introduce a new stochastic Exner model and discuss it using numerical simulations in an appropriate test case. (10.1051/proc/201448015)
DOI : 10.1051/proc/201448015
A new model for shallow viscoelastic free-surface flows forced by gravity on rough inclined bottom
- Boyaval Sébastien
ESAIM: Proceedings, EDP Sciences , 2014, 45, pp.108 - 117 . A 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.