Imperial College London

ProfessorPierreDegond

Faculty of Natural SciencesDepartment of Mathematics

Chair in Applied Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 1474p.degond Website CV

 
 
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Location

 

6M38Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

292 results found

Degond PAA, Deluzet F, 2017, Asymptotic-Preserving methods and multiscale models for plasma physics, Journal of Computational Physics, Vol: 336, Pages: 429-457, ISSN: 0021-9991

The purpose of the present paper is to provide an overview of Asymptotic-Preserving methods for multiscale plasma simulations by addressing three sin-gular perturbation problems. First, the quasi-neutral limit of fluid and kineticmodels is investigated in the framework of non magnetized aswell as magne-tized plasmas. Second, the drift limit for fluid descriptions of thermal plasmasunder large magnetic fields is addressed. Finally efficient numerical resolutionsof anisotropic elliptic or diffusion equations arising in magnetized plasma simu-lation are reviewed.

Journal article

Degond PAA, Manhart A, Yu H, 2017, A continuum model for nematic alignment of self-propelled particles, Discrete and Continuous Dynamical Systems - Series B, Vol: 22, Pages: 1295-1327, ISSN: 1553-524X

A continuum model for a population of self-propelled particlesinteracting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematicmean direction of the ow and the densities of particles owing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.

Journal article

Chertock A, Degond PAA, Neusser J, 2017, An asymptotic-preserving method for a relaxation of the NavierStokes-Korteweg equations, Journal of Computational Physics, Vol: 335, Pages: 387-4003, ISSN: 0021-9991

The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interfacemodel for compressible two-phase flows. As direct numericalsimulations basedon the NSK system are quite expensive and in some cases even impossible, weconsider a relaxation of the NSK system, for which robust numerical methodscan be designed. However, time steps for explicit numericalschemes depend onthe relaxation parameter and therefore numerical simulations in the relaxationlimit are very inefficient. To overcome this restriction, we propose an implicit-explicit asymptotic-preserving finite volume method. We prove that the newscheme provides a consistent discretization of the NSK system in the relaxationlimit and demonstrate that it is capable of accurately and efficiently computingnumerical solutions of problems with realistic density ratios and small interfacialwidths.

Journal article

Degond P, Liu J-G, Merino-Aceituno S, Tardiveau Tet al., 2017, Continuum dynamics of the intention field under weakly cohesive social interaction, Mathematical Models and Methods in Applied Sciences, Vol: 27, ISSN: 1793-6314

We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, we study conditions under which the Fokker–Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.

Journal article

Degond PAA, Peurichard D, 2017, Modelling tissue self-organization: from micro to macro models, Lecture Notes in Computational Science and Engineering, Vol: 122, Pages: 93-108, ISSN: 1439-7358

In this chapter, we present recent works concerned with the derivation of a macroscopic model for complex interconnected fiber networks from an agent-based model, with applications to, but not limited to, adipose tissue self-organization. Starting from an agent-based model for interconnected fibers interacting through alignment interactions and having the ability to create and suppress cross-links, the formal limit of large number of individuals is first investigated. It leads to a kinetic system of two equations: one for the individual fiber distribution function and one for the distribution function of connected fiber pairs. The hydrodynamic limit, in a regime of instantaneous fiber linking/unlinking then leads to a macroscopic model describing the evolution of the fiber local density and mean orientation. These works are the first attempt to derive a macroscopic model for interconnected fibers from an agent-based formulation and represent a first step towards the formulation of a large scale synthetic tissue model which will serve for the investigation of large scale effects in tissue homeostasis.

Journal article

Degond PAA, Ferreira MA, Motsch S, 2016, Damped Arrow-Hurwicz algorithm for sphere packing, Journal of Computational Physics, Vol: 332, Pages: 47-65, ISSN: 1090-2716

We consider algorithms that, from an arbitrarily sampling ofNspheres(possibly overlapping), nd a close packed con guration without overlapping.These problems can be formulated as minimization problems with non-convexconstraints. For such packing problems, we observe that the classical iterativeArrow-Hurwicz algorithm does not converge. We derive a novel algorithmfrom a multi-step variant of the Arrow-Hurwicz scheme with damping. Wecompare this algorithm with classical algorithms belonging to the class oflinearly constrained Lagrangian methods and show that it performs better.We provide an analysis of the convergence of these algorithms in the simplecase of two spheres in one spatial dimension. Finally, we investigate thebehaviour of our algorithm when the number of spheres is large in two andthree spatial dimensions.

Journal article

Degond PAA, Deluzet F, Doyen D, 2016, Asymptotic-Preserving Particle-In-Cell methods for the Vlasov-Maxwell system in the quasi-neutral limit, Journal of Computational Physics, Vol: 330, Pages: 467-492, ISSN: 0021-9991

In this article, we design Asymptotic-Preserving Particle-In-Cell methods for the Vlasov-Maxwell system inthe quasi-neutral limit, this limit being characterized by a Debye length negligible compared to the space scaleof the problem. These methods are consistent discretizations of the Vlasov-Maxwell system which, in thequasi-neutral limit, remain stable and are consistent with a quasi-neutral model (in this quasi-neutral model,the electric eld is computed by means of a generalized Ohm law). The derivation of Asymptotic-Preservingmethods is not straightforward since the quasi-neutral model is a singular limit of the Vlasov-Maxwell model.The key step is a reformulation of the Vlasov-Maxwell system which uni es the two models in a single setof equations with a smooth transition from one to another. As demonstrated in various and demandingnumerical simulations, the Asymptotic-Preserving methods are able to treat efficiently both quasi-neutralplasmas and non-neutral plasmas, making them particularly well suited for complex problems involving denseplasmas with localized non-neutral regions.

Journal article

Degond PAA, Henkes S, Yu H, 2016, Self-organized Hydrodynamics with density-dependent velocity, Kinetic and Related Models, Vol: 10, Pages: 193-213, ISSN: 1937-5093

Motivated by recent experimental and computational results thatshow a motility-induced clustering transition in self-propelled particle systems,we study an individual model and its corresponding Self-Organized Hydrodynamicmodel for collective behaviour that incorporates a density-dependentvelocity, as well as inter-particle alignment. The modal analysis of the hydrodynamicmodel elucidates the relationship between the stability of the equilibriaand the changing velocity, and the formation of clusters. We find, in agreementwith earlier results for non-aligning particles, that the key criterion for stabilityis (ρv(ρ))0 ≥ 0, i.e. a nondecreasing mass flux ρv(ρ) with respect to the density.Numerical simulation for both the individual and hydrodynamic modelswith a velocity function inspired by experiment demonstrates the validity ofthe theoretical results.

Journal article

Degond PAA, Filbet F, 2016, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field : formal derivation., Journal of Statistical Physics, Vol: 165, Pages: 765-784, ISSN: 0022-4715

In this paper we establish the asymptotic limit of the three dimensional Vlasov–Poissonequation with strong external magnetic field. The guiding center approximation is investigated inthe three dimensional case with a non-constant magnetic field. In the long time asymptotic limit,the motion can be split in two parts : one stationary flow along the lines of the magnetic fieldand the guiding center motion in the orthogonal plane of the magnetic field where classical driftvelocities and invariants (magnetic moment) are recovered.

Journal article

Creppy A, Plouraboué F, Praud O, Druart X, Cazin S, Yu H, Degond PAAet al., 2016, Symmetry-breaking phase-transitions in highly concentrated semen, Journal of the Royal Society Interface, Vol: 13, ISSN: 1742-5689

New experimental evidence of self-motion of aconfined active suspension is presented. Depositingfresh semen sample in an annular shaped micro-fluidic chip leads to a spontaneous vortex state ofthe fluid at sufficiently large sperm concentration.The rotation occurs unpredictably clockwise orcounterclockwise and is robust and stable. Furthermore,for highly active and concentrated semen, richerdynamics can occur such as self-sustained or dampedrotation oscillations. Experimental results obtainedwith systematic dilution provide a clear evidence of aphase transition toward collective motion associatedwith local alignment of spermatozoa akin to theVicsek model. A macroscopic theory based onpreviously derived Self-Organized Hydrodynamics(SOH) models is adapted to this context and providespredictions consistent with the observed stationarymotion.

Journal article

Degond P, Liu J-G, Pego RL, 2016, Coagulation-fragmentation model for animal group-size statistics, Journal of Nonlinear Science, Vol: 27, Pages: 379-424, ISSN: 1432-1467

We study coagulation-fragmentation equations inspired by a simple modelproposed in fisheries science to explain data for the size distribution ofschools of pelagic fish. Although the equations lack detailed balance and admitno $H$-theorem, we are able to develop a rather complete description ofequilibrium profiles and large-time behavior, based on recent developments incomplex function theory for Bernstein and Pick functions. In thelarge-population continuum limit, a scaling-invariant regime is reached inwhich all equilibria are determined by a single scaling profile. This universalprofile exhibits power-law behavior crossing over from exponent $-\frac23$ forsmall size to $-\frac32$ for large size, with an exponential cut-off.

Journal article

Barbaro ABT, Canizo JA, Carrillo de la plata J, Degond PAAet al., 2016, Phase Transitions in a Kinetic Flocking Model of Cucker-Smale Type, Multiscale Modeling & Simulation, Vol: 14, Pages: 1063-1088, ISSN: 1540-3467

We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a “disordered” to an “ordered” state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.

Journal article

Blanchet A, Degond PAA, 2016, Topological interactions in a Boltzmann-type framework, Journal of Statistical Physics, Vol: 163, Pages: 41-60, ISSN: 1572-9613

We consider a finite number of particles characterised by their positionsand velocities. At random times a randomly chosen particle, the follower,adopts the velocity of another particle, the leader. The follower choosesits leader according to the proximity rank of the latter with respect to theformer. We study the limit of a system size going to infinity and, under theassumption of propagation of chaos, show that the limit equation is akinto the Boltzmann equation. However, it exhibits a spatial non-localityinstead of the classical non-locality in velocity space. This result relies onthe approximation properties of Bernstein polynomials. We illustrate thedynamics with numerical simulations.

Journal article

Degond P, Delebecque F, Peurichard D, 2015, Continuum model for linked fibers with alignment interactions, Mathematical Models and Methods in Applied Sciences, Vol: 26, ISSN: 0218-2025

We introduce an individual-based model for fiber elements having the abilityto cross-link or unlink each other and to align with each other at the cross links.We first formally derive a kinetic model for the fiber and cross-links distributionfunctions. We then consider the fast linking/unlinking regime in which the modelcan be reduced to the fiber distribution function only and investigate its diffusionlimit. The resulting macroscopic model consists of a system of nonlinear diffusionequations for the fiber density and mean orientation. In the case of a homogeneousfiber density, we show that the model is elliptic.

Journal article

Muljadi B, Narski J, Lozinski A, Degond Pet al., 2015, Non-conforming multiscale finite element method for stokes flows in heterogeneous media. Part I: Methodologies and numerical experiments, Multiscale Modeling & Simulation, Vol: 13, Pages: 1146-1172, ISSN: 1540-3467

The Multiscale Finite Element Method (MsFEM) is developed in the vein of Crouzeix-Raviart element for solving viscous incompressible ows in genuine heterogeneous media. Such owsare relevant in many branches of engineering, often at multiple scales and at regions where analyticalrepresentations of the microscopic features of the ows are often unavailable. Full accounts to theseproblems heavily depend on the geometry of the system under consideration and are computationallyexpensive. Therefore, a method capable of solving multiscale features of the ow without con ningitself to ne scale calculations is sought after.The approximation of boundary condition on coarse element edges when computing the multiscalebasis functions critically inuences the eventual accuracy of any MsFEM approaches. The weaklyenforced continuity of Crouzeix - Raviart function space across element edges leads to a naturalboundary condition for the multiscale basis functions which relaxes the sensitivity of our method tocomplex patterns of obstacles exempt from the needs of implementing any oversampling techniques.Additionally, the application of penalization method makes it possible to avoid complex unstructureddomain and allows extensive use of simpler Cartesian meshes.Key words. Crouzeix-Raviart Element, Multiscale Finite Element Method, Stokes Equations,Penalization Method

Journal article

Degond P, Fibich G, Piccoli B, Tadmor Eet al., 2015, SPECIAL ISSUE ON MODELING AND CONTROL IN SOCIAL DYNAMICS, NETWORKS AND HETEROGENEOUS MEDIA, Vol: 10, Pages: I-II, ISSN: 1556-1801

Journal article

Degond P, Navoret L, 2015, A multi-layer model for self-propelled disks interacting through alignment and volume exclusion, Mathematical Models and Methods in Applied Science, Vol: 25, Pages: 2439-2475, ISSN: 0218-2025

We present an individual-based model describing disk-like self-propelled particles movinginside parallel planes. The disk directions of motion follow alignment rules insideeach layer. Additionally, the disks are subject to interactions with those of the neighboringlayers arising from volume exclusion constraints. These interactions affect the diskinclinations with respect to the plane of motion. We formally derive a macroscopic modelcomposed of planar self-organized hydrodynamic (SOH) models describing the transportof mass and evolution of mean direction of motion of the disks in each plane, supplementedwith transport equations for the mean disk inclination. These planar modelsare coupled due to the interactions with the neighboring planes. Numerical comparisonsbetween the individual-based and macroscopic models are carried out. These modelscould be applicable, for instance, to describe sperm-cell collective dynamics.

Journal article

David I, Kohnke P, Lagriffoul G, Praud O, Plouarboué F, Degond P, Druart Xet al., 2015, Mass sperm motility is associated with fertility in sheep, Animal Reproduction Science, Vol: 161, Pages: 75-81, ISSN: 0378-4320

The study was to focus on the relationship between wave motion (mass sperm motility, measured by a mass sperm motility score, manually assessed by artificial insemination (AI) center operators) and fertility in male sheep. A dataset of 711,562 artificial inseminations performed in seven breeds by five French AI centers during the 2001 to 2005 time period was used for the analysis. Factors influencing the outcome of the insemination, which is a binary response observed at lambing of either success (1) or failure (0), were studied using a joint model within each breed and AI center (eight separate analyses). The joint model is a multivariate model where all information related to the female, the male and the insemination process were included to improve the estimation of the factor effects. Results were consistent for all analyses. The male factors affecting AI results were the age of the ram and the mass motility. After correction for the other factors of variation, the lambing rate increased quasi linearly from three to more than ten points with the mass sperm motility score depending on the breed and the AI center. The consistency of the relationship for all breeds indicated that mass sperm motility is predictive of the fertility resulting when sperm are used from a specific ejaculate. Nonetheless, predictability could be improved if an objective measurement of mass sperm motility were available as a substitute for the subjective scoring currently in use in AI centers.

Journal article

Fehrenbach J, Narski J, Hua J, Lemercier S, Jelic A, Appert-Rolland C, Donikian S, Pettre J, Degond Pet al., 2015, Time-delayed follow-the-leader model for pedestrians walking in line, Networks and Heterogeneous Media, Vol: 10, Pages: 579-608, ISSN: 1556-181X

We use the results of a pedestrian tracking experiment to identify a follow-the-leader model for pedestrians walking-in-line. We demonstrate the existence of a time-delay between a subject's response and the predecessor's corresponding behavior. This time-delay induces an instability which can be damped out by a suitable relaxation. By comparisons with the experimental data, we show that the model reproduces well the emergence of large-scale structures such as congestions waves. The resulting model can be used either for modeling pedestrian queuing behavior or can be incorporated into bi-dimensional models of pedestrian traffic.

Journal article

Carlen E, Carvalho MC, Degond P, Wennberg Bet al., 2015, A Boltzmann model for rod alignment and schooling fish, Nonlinearity, Vol: 28, Pages: 1783-1803, ISSN: 1361-6544

We consider a Boltzmann model introduced by Bertin, Droz and Grégoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The analysis is carried over rigorously when there are only finitely many non-zero Fourier modes of the noise distribution. In this case, we can show that the critical exponent of the bifurcation is exactly 1/2. In the case of an infinite number of non-zero Fourier modes, a similar behavior can be formally obtained thanks to a method relying on integer partitions first proposed by Ben-Naïm and Krapivsky.

Journal article

Degond P, Lozinski A, Muljadi BP, Narski Jet al., 2015, Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media, Communications in Computational Physics, Vol: 17, Pages: 887-907, ISSN: 1991-7120

The adaptation of Crouzeix-Raviart finite element in the context of multi-scale finite element method (MsFEM) is studied and implemented on diffusion and advection-diffusion problems in perforated media. It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.

Journal article

Degond P, Yu H, 2015, Self-organized hydrodynamics in an annular domain: Modal analysis and nonlinear effects, MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, Vol: 25, Pages: 495-519, ISSN: 0218-2025

Journal article

Degond P, Dimarco G, Thi BNM, Wang Net al., 2015, Macroscopic models of collective motion with repulsion, Communications in Mathematical Sciences, Vol: 13, Pages: 1615-1638, ISSN: 1945-0796

Journal article

Degond P, Liu J-G, Ringhofer C, 2014, Evolution of wealth in a non-conservative economy driven by local Nash equilibria, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 372, ISSN: 1364-503X

Journal article

Degond P, Frouvelle A, Liu J-G, 2014, Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics, Archive for Rational Mechanics and Analysis, Vol: 216, Pages: 63-115, ISSN: 1432-0673

We provide a complete and rigorous description of phase transitions for kineticmodels of self-propelled particles interacting through alignment. These modelsexhibit a competition between alignment and noise. Both the alignment frequencyand noise intensity depend on a measure of the local alignment. We show that, inthe spatially homogeneous case, the phase transition features (number and nature ofequilibria, stability, convergence rate, phase diagram, hysteresis) are totally encodedin how the ratio between the alignment and noise intensities depend on the localalignment. In the spatially inhomogeneous case, we derive the macroscopic modelsassociated to the stable equilibria and classify their hyperbolicity according to thesame function.

Journal article

Degond P, Frouvelle A, Raoul G, 2014, Local Stability of Perfect Alignment for a Spatially Homogeneous Kinetic Model, JOURNAL OF STATISTICAL PHYSICS, Vol: 157, Pages: 84-112, ISSN: 0022-4715

Journal article

Barbaro ABT, Degond P, 2014, PHASE TRANSITION AND DIFFUSION AMONG SOCIALLY INTERACTING SELF-PROPELLED AGENTS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, Vol: 19, Pages: 1249-1278, ISSN: 1531-3492

Journal article

Chae D, Degond P, Liu J-G, 2014, Well-posedness for Hall-magnetohydrodynamics, ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, Vol: 31, Pages: 555-565, ISSN: 0294-1449

Journal article

Degond P, Liu J-G, Ringhofer C, 2014, Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria, JOURNAL OF NONLINEAR SCIENCE, Vol: 24, Pages: 93-115, ISSN: 0938-8974

Journal article

Degond P, Dimarco G, Thi BNM, 2014, HYDRODYNAMICS OF THE KURAMOTO-VICSEK MODEL OF ROTATING SELF-PROPELLED PARTICLES, MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, Vol: 24, ISSN: 0218-2025

Journal article

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