We consider multi-dimensional extensions of Maxwell’s seminal rheological 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 propagate 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.

We extend our analysis on the Oldroyd-B model in Barrett & Boyaval (2011, Math. Models Methods Appl. Sci.,21, 1783–1837) to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain |$\mathscr{D} \subset {\mathbb R}^d$|⁠, |$d=2$| or |$3$|⁠, subject to no-flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation 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 discretization, 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.

Selecting appropriate performance measures is crucial for calibrating numerical flood models with satellite images, as differences between performance measures can significantly impact model predictions. The present study reports a step‐by‐step methodology to guide the selection of performance measures to compare numerical simulations of a flood event with a satellite image, for purposes of model calibration or validation. Faced with a set of performance measures, practitioners need to assess which are most suitable for comparing observed and simulated flood maps. First, a correlation analysis was conducted to determine the relationships between various performance measures. A series of criteria were then applied to evaluate the sensitivity of the performance measures with respect to flood extent, satellite characteristics (position and orientation), and errors in measurement or processing (raw satellite values or extraction of flood maps). The performance measures meeting the most criteria were then ranked based on the identifiability of parameter values with regard to observation errors. The methodology is discussed numerically in the context of a single flood event (on the Garonne River in France in February 2021), using a distribution of water depths simulated from a shallow‐water model (parameterized by an uncertain friction field). The methodology was shown to be useful in the study case in identifying 4 performance measures out of 24. The proposed methodology is intended to be generic, and therefore, applicable to any flood model or flood event.

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 regularities 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. Résumé. Nous effectuons une étude numérique de l'équation de Burgers non visqueuse en dimension un d'espace, avec des conditions aux limites périodiques et un terme source stochastique de moyenne spatiale nulle. Ce terme source possède la régularité d'un bruit blanc en temps, tandis que nous considérons différentes régularités en espace. Pour les tests numériques, nous utilisons un schéma de volumes finis combinant une intégration en temps de type Euler-Maruyama avec le flux numérique de Godunov. Nous effectuons des simulations avec la méthode de Monte-Carlo et analysons les résultats pour différentes régularités en accordant une attention particulière au comportement en temps long. Il apparaît que la moyenne empirique des réalisations converge toujours vers la moyenne en espace de la condition initiale (déterministe) quand t → ∞, comme c'est le cas pour la solution du problème sans terme source, même dans le cas où le terme stochastique est peu régulier en espace. Par ailleurs, la variance empirique converge elle aussi en temps long, vers une valeur qui dépend de la régularité et de l'amplitude du terme stochastique.