Convective self-aggregation is the spontaneous spatial organisation of deep-convective clouds into a limited region surrounded by drier, convectively inhibited regions, which occurs in many models. We propose a simple, piecewise linear model for self-aggregation based on primitive equations. In this model, each atmospheric column is in one of two possible thermodynamic regimes, deep convective or convectively inhibited, and the thermodynamics in each regime is linearised. The model simulates aggregated and non-aggregated steady states and reproduces many properties of self-aggregation as simulated by kilometre-resolution models. In particular, it exhibits a hysteresis with multiple, aggregated and non-aggregated equilibria and a similar sensitivity to convective inhibition, domain size, and boundary-layer radiative cooling in the dry region as in kilometre-resolution simulations. These results suggest that a self-aggregated state can be considered as a simple gravity wave with two phases: one convective and one convectively inhibited.

We present and study a Particle method for the stationary solutions of a class of transport equations. This method is inspired by non-stationary Particle methods, the time variable being replaced by one spatial variable. Particles trajectories are computed using the ``time-dependent'' equations, and then the approximation is based on a quadrature method using the particle locations as quadrature points. We prove the convergence of the scheme under suitable regularity assumptions on the data and the solution, together with a ``characteristic completeness'' assumption (the characteristic curves fullfill the whole computational domain). We also provide an error estimate. The scheme is tested numerically on a two dimensional linear equation and we present a numerical study of convergence. Finally, we use this method to carry out numerical simulations of a landscape evolution model, where an erodible topography evolves under the effects of water erosion and sedimentation. The scheme is then useful to deal with wet/dry areas.

In this article, we consider the general task of performing Gaussian process regression (GPR) on pointwise observations of solutions of the 3 dimensional homogeneous free space wave equation. In a recent article, we obtained promising covariance expressions tailored to this equation: we now explore the potential applications of these formulas. We first study the particular cases of stationarity and radial symmetry, for which significant simplifications arise. We next show that the true-angle multilateration method for point source localization, as used in GPS systems, is naturally recovered by our GPR formulas in the limit of the small source radius. Additionally, we show that this GPR framework provides a new answer to the ill-posed inverse problem of reconstructing initial conditions for the wave equation from a limited number of sensors, and simultaneously enables the inference of physical parameters from these data. We finish by illustrating this "physics informed" GPR on a number of practical examples.

In this paper, we study a system of partial differential equations modeling the evolution of a landscape in order to describe the mechanisms of pattern formations. A ground surface is eroded by the flow of water over it, by either sedimentation or dilution. We consider a model, composed of three evolution equations: one on the elevation of the ground surface, one on the fluid height and one on the concentration of sediments in the fluid layer. We first establish the well-posedness of the system in short time and under the assumption that the initial fluid height does not vanish. Then, we focus on pattern formation in the case of a film flow over an inclined erodible plane. For that purpose, we carry out a spectral stability analysis of constant state solutions in order to determine instability conditions and identify a mechanism for pattern formations. These patterns, which are rills and gullies, are the starting point of the formation of rivers and valleys in landscapes. Finally, we carry out some numerical simulations of the full system in order to validate the spectral instability scenario, and determine the resulting patterns.

The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler-Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual spectral analysis and, beyond specific computations, newly devised arguments. The growth rate with respect to the amplitude of the background waves is also provided when the instability occurs.

Let $L$ be a linear differential operator acting on functions defined over an open set $\mathcal{D}\subset \mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{x\in\mathcal{D}}$ whose sample paths all verify the partial differential equation (PDE) $L(u) = 0$, solely in terms of their first two moments. When compared to previous similar results, the novelty lies in that the equality $L(u) = 0$ is understood in the sense of distributions, which is a powerful functional analysis framework mostly designed to study linear PDEs. This framework enables to reduce to the minimum the required differentiability assumptions over the first two moments of $(U(x))_{x\in\mathcal{D}}$ as well as over its sample paths in order to make sense of the PDE $L(U_{\omega})=0$. In view of Gaussian process regression (GPR) applications, we show that when $(U(x))_{x\in\mathcal{D}}$ is a Gaussian process (GP), the sample paths of $(U(x))_{x\in\mathcal{D}}$ conditioned on pointwise observations still verify the constraint $L(u)=0$ in the distributional sense. We finish by deriving a simple but instructive example, a GP model for the 3D linear wave equation, for which our theorem is applicable and where the previous results from the literature do not apply in general.

Absorbing boundary conditions are important when one simulates the propagation of waves on a bounded numerical domain without creating artificial reflections. In this paper, we consider various hyperbolic–dispersive equations modeling water wave propagation. A typical example is the Korteweg–de Vries equation 1 u_t+u u_x+ε u_xxx=0, ∀ x∈ℝ, ∀ t>0. In the case of linearized equations, some progress was recently done for one dimensional scalar dispersive equations using discrete transparent boundary conditions. However, a generalization of this approach to multi-dimensional setting is not obvious. In this paper, we consider the alternative perfectly matched layer (PML) approach for the linearized Korteweg–de Vries equation: 2 u_t+U u_x+ε u_xxx=0 ∀ x∈ℝ, ∀ t>0, where U∈ℝ denotes a reference speed. We first propose a direct perfectly matched layer approach and study the stability of the modified system. These equations are not always stable, the main obstruction being the classical condition v_g(k)v_ϕ (k)≥ 0 found in the literature on PML (Bécache et al in J Comput Phys 188(2):399–433, 2003) that we recover in our analysis. Then, we introduce a hyperbolic system with a source term that is an approximation of the Korteweg–de Vries equations. In this case, the complete PML equations are not, again, completely stable. However, a version of the PML equations for this system derived without the source term is found to be stable and can absorb outgoing waves although it may create reflections as it is not perfectly matched. Finally, we consider the BBM–Boussinesq system that models bi-directional waves at the surface of an inviscid fluid layer. The dispersive properties for a subclass of physically relevant models are better suited for PML techniques since the condition v_g(k)v_ϕ (k)≥ 0 is always satisfied. We show that the PML equations are always stable in this case. We illustrate numerically the absorbing and stability properties of these PML models and provide also KdV type simulation by choosing properly initial data.

In this paper, we present a general framework to construct section-averaged models when the flow is constrained – e.g. by topography – to be almost one-dimensional. These models are consistent with the two-dimensional shallow water equations. After rewriting the two-dimensional shallow water equations in a suitable set of coordinates allowing to take care of a meandering configuration, we consider the quasi one-dimensional regime. Then, we expand the water elevation and velocity field in the spirit of the diffusive wave equations and establish a set of one-dimensional equations made of a mass, momentum and energy equations, which are close to the ones usually used in hydraulic engineering. Our model reduces to classical shallow water models with variable sections found in the literature. Out of these configurations, there is an O(1) deviation of our model from the classical ones. Finally, we present the main mathematical properties of our model and carry out numerical simulations to validate our approach by comparing the results to the full two-dimensional shallow water equations.

We develop a general strategy in order to implement approximate discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leap-frog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle is analyzed by means of the so-called normal mode analysis. Numerical investigations for the full problem on the rectangle show that strong instabilities may occur when coupling stable strategies on each side of the rectangle. Other coupling strategies yield promising results.

Let P be a linear differential operator over D Ă R and U “ pUxqxPD a second order stochastic process. In the first part of this article, we prove a new necessary and sufficient condition for all the trajectories of U to verify the partial differential equation (PDE) T pUq “ 0. This condition is formulated in terms of the covariance kernel of U . When compared to previous similar results [1], the novelty of this result is that the equality T pUq “ 0 is understood in the sense of distributions, which is a functional analysis framework particularly adapted to the study of PDEs. This theorem provides precious insights during the second part of this article, which is dedicated to performing ”physically informed” machine learning on data that is solution to the homogeneous 3 dimensional free space wave equation. We perform Gaussian process regression (GPR) on this data, which is a kernel based Bayesian approach to machine learning. To do so, we put Gaussian process (GP) priors over the wave equation’s initial conditions and propagate them through the wave equation. We obtain explicit formulas for the covariance kernel of the posterior GP; this kernel can then be used for GPR. Our theorem states that this kernel, the trajectories of the corresponding GP and the predictions provided by GPR are all solutions to the wave equation in the sense of distributions. We explore two particular cases : the radial symmetry and the point source. For the former, we derive convolution-free GPR formulas; for the latter, we show a direct link between GPR and the classical triangulation method for point source localization used e.g. in GPS systems. Additionally, this Bayesian framework gives rise to a new answer for the ill-posed inverse problem of reconstructing initial conditions for the wave equation with finite dimensional data, and simultaneously provides a way of estimating physical parameters from this data as in [2]. We finish by showcasing this physically informed GPR on a number of practical examples.

In this paper, we introduce artificial boundary conditions for the linearized Green-Naghdi system of equations. The derivation of such continuous (respectively discrete) boundary conditions include the inversion of Laplace transform (respectively Z-transform) and these boundary conditions are in turn non local in time. In the case of continuous boundary conditions, the inversion is done explicitly. We consider two spatial discretisations of the initial system either on a staggered grid or on a collocated grids, both of interest from the practical point of view. We use a Crank Nicolson time discretization. The proposed numerical scheme with the staggered grid permits explicit Z-transform inversion whereas the collocated grid discretization do not. A stable numerical procedure is proposed for this latter inversion. We test numerically the accuracy of the described method with standard Gaussian initial data and wave packet initial data which are more convenient to explore the dispersive properties of the initial set of equations. We used our transparent boundary conditions to solve numerically the problem of injecting propagating (planar) waves in a computational domain.

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.

In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch andB. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF) dissipative entropyintroduced to study the lubrications equations. Dierent dissipative entropies are obtained playing with the dragterms on the viscous shallow water equations. It helps for instance to prove global existence of nonnegativeweak solutions for the lubrication equations starting from the global existence of nonnegative weak solutions forappropriate viscous shallow-water equations.

We carry out a systematic analytical and numerical study of spectral stability of discontinuous roll wave solutions of the inviscid Saint-Venant equations, based on a periodic Evans–Lopatinsky determinant analogous to the periodic Evans function of Gardner in the (smooth) viscous case, obtaining a complete spectral stability diagram useful in hydraulic engineering and related applications. In particular, we obtain an explicit low-frequency stability boundary, which, moreover, matches closely with its (numerically-determined) counterpart in the viscous case. This is seen to be related to but not implied by the associated formal first-order Whitham modulation equations.

We consider various approximations of artificial boundary conditions for linearized Benjamin–Bona–Mahony BBM equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the \(\mathcal {Z}\)-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We focus on and prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our transparent boundary conditions.

This paper provides the full proof of the results announced by the authors in [C. R. Acad. Sciences (2016)]. We introduce an original relative entropy for compressible Navier-Stokes equations with density dependent viscosities and discuss some possible applications such as inviscid limit or low Mach number limit. We first consider the case µ(ϱ) = µϱ and λ(ϱ) = 0 and a pressure law under the form p(ϱ) = aϱγ with γ > 1, which corresponds in particular to the formulation of the viscous shallow water equations. We present some mathematical results related to the weak-strong uniqueness, the convergence to a dissipative solution of compressible or incompressible Euler equations. Moreover, we show the convergence of the viscous shallow water equations to the inviscid shallow water equations in the vanishing viscosity limit and further prove convergence to the incompressible Euler system in the low Mach limit. This extends results with constant viscosities recently initiated by E. Feireisl, B.J. Jin and A. Novotny in [J. Math. Fluid Mech. (2012)].

We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude number or “weakly unstable” limit \(F\rightarrow 2^+\) and for general values of the Froude number F, including the limit \(F\rightarrow +\infty \). In the former, \(F\rightarrow 2^+\), limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto–Sivashinsky (KdV–KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as \(F\rightarrow 2^+\) is equivalent to stability of the corresponding KdV–KS waves in the KdV limit. Together with recent results obtained for KdV–KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around \(F=2.3\) from weakly unstable to different, large-F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically \(2.5\le F\le 6.0\).

In this paper we present a general framework to construct 1D width averaged models when the flow is constrained -e.g. by topography- to be almost 1D. We start from two dimensional shallow water equations, perform an asymptotic expansion of the fluid elevation and velocity field in the spirit of wave diffusive equations and establish a set of 1D equations made of a mass, momentum and energy equations which are close to the one usually used in hydraulic engineering. We show that in some special cases, like the U-shaped river bed, that our set of equations reduces to the classical 1d shallow water equations. Out of these configurations, there is an O (1) deviation of our model from the classical one.