Reflective Conditions for Radiative Transfer in Integral Form with H-Matrices

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a $P^1$ Finite Element Method. The convolution integrals are precomputed at every vertices of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is $O(N\sqrt[3]{N}\ln N)$, with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex

Simulation of the 3D Radiative Transfer with Anisotropic Scattering for Convective Trails

The integro-differential formulation of the RTE and its solution by iterations on the source has been extended here to handle anisotropic scattering. The iterative part of the method is O(N ln N ), thanks to an efficient use of H-matrices. The precision is good enough to evaluate the effect of sensitive parameters for the study of contrails. Most of the time the stratified 1D approximation should suffice, but in complex cases with high relief the 3D formulation is needed

Absorption de l'eau et des nutriments par les racines des plantes : modélisation, analyse et simulation

In the context of the development of sustainable agriculture aiming at preserving natural resources and ecosystems, it is necessary to improve our understanding of underground processes and interactions between soil and plant roots.In this thesis, we use mathematical and numerical tools to develop explicit mechanistic models of soil water and solute movement accounting for root water and nutrient uptake and governed by nonlinear partial differential equations. An emphasis is put on resolving the geometry of the root system as well as small scale processes occurring in the rhizosphere, which play a major role in plant root uptake.The first study is dedicated to the mathematical analysis of a model of phosphorus (P) uptake by plant roots. The evolution of the concentration of P in the soil solution is governed by a convection-diffusion equation with a nonlinear boundary condition at the root surface, which is included as a boundary of the soil domain. A shape optimization problem is formulated that aims at finding root shapes maximizing P uptake.The second part of this thesis shows how we can take advantage of the recent advances of scientific computing in the field of unstructured mesh adaptation and parallel computing to develop numerical models of soil water and solute movement with root water and nutrient uptake at the plant scale while taking into account local processes at the single root scale.Dans le contexte du développement d'une agriculture durable visant à préserver les ressources naturelles et les écosystèmes, il s'avère nécessaire d'approfondir notre compréhension des processus souterrains et des interactions entre le sol et les racines des plantes.Dans cette thèse, on utilise des outils mathématiques et numériques pour développer des modèles mécanistiques explicites du mouvement de l'eau et des nutriments dans le sol et de l'absorption racinaire, gouvernés par des équations aux dérivées partielles non linéaires. Un accent est mis sur la prise en compte explicite de la géométrie du système racinaire et des processus à petite échelle survenant dans la rhizosphère, qui jouent un rôle majeur dans l'absorption racinaire.La première étude est dédiée à l'analyse mathématique d'un modèle d'absorption du phosphore (P) par les racines des plantes. L'évolution de la concentration de P dans la solution du sol est gouvernée par une équation de convection-diffusion avec une condition aux limites non linéaire à la surface de la racine, que l'on considère ici comme un bord du domaine du sol. On formule ensuite un problème d'optimisation de forme visant à trouver les formes racinaires qui maximisent l'absorption de P.La seconde partie de cette thèse montre comment on peut tirer avantage des récents progrès du calcul scientifique dans le domaine de l'adaptation de maillage non structuré et du calcul parallèle afin de développer des modèles numériques du mouvement de l'eau et des solutés et de l'absorption racinaire à l'échelle de la plante, tout en prenant en compte les phénomènes locaux survenant à l'échelle de la racine unique

Discrete nonlinear Schrödinger equations for periodic optical systems : pattern formation in \chi(3) coupled waveguide arrays

Discrete nonlinear Schrödinger equations have been used for many years to model the propagation of light in optical architectures whose refractive index profile is modulated periodically in the transverse direction. Typically, one considers a modal decomposition of the electric field where the complex amplitudes satisfy a coupled system that accommodates nearest neighbour linear interactions and a local intensity dependent term whose origin lies in the χ (3) contribution to the medium's dielectric response. In this presentation, two classic continuum configurations are discretized in ways that have received little attention in the literature: the ring cavity and counterpropagating waves. Both of these systems are defined by distinct types of boundary condition. Moreover, they are susceptible to spatial instabilities that are ultimately responsible for generating spontaneous patterns from arbitrarily small background disturbances. Good agreement between analytical predictions and simulations will be demonstrated

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methods—i.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methods—i.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%

A δf\delta f PIC method with auxiliary Forward-Backward Lagrangian reconstructions

In this note we describe a $\delta f$ particle method where the bulk density is periodically remapped on a coarse spline grid using a Forward-Backward Lagrangian (FBL) approach. We describe the method in the case of an electrostatic PIC scheme and validate its qualitative properties using a classical two-stream instability subject to a uniform oscillating drive