Programma - Dipartimento di Matematica

Numerical Aspects of Hyperbolic Balance Laws and Related Problems
Monday
14.15-14.55 Second-order Euler-preserving wall condition for the BGK equation on
Cartesian meshes
F. Bernard, Politecnico di Torino
A typical feature of complex flows is the presence of different flow regime (kinetic and continuum) in the same field. Thus the kinetic scheme has to be Asymptotic Preserving (AP) to
ensure a smooth transition between the two regimes. Dealing with immersed bodies implies to
preserve also these properties up to the bodies. Here, a simple second order scheme on Cartesian grids for the BGK equation is presented, with emphasis on the enforcement of boundary
conditions on immersed bodies. The continuum limit of the BGK model is studied. A new way
to enforce the boundary condition preserving the asymptotic limit towards Euler equations up
to the body is presented. This property ensures a smooth transition towards the hydrodynamic regime. Applications to one-dimensional and two-dimensional test cases are presented to
illustrate the accuracy of the method.
14.55-15.35 Schemi Runge-Kutta multirate per ODE e leggi di conservazione
G. Visconti, M. Semplice, Università di Torino
Gli schemi Runge-Kutta Multirate sono stati sviluppati per trattare i sistemi di Equazioni
Differenziali Ordinarie in cui vi sia un naturale partizionamento delle variabili fra componenti
attive e componenti lente. Essi migliorano l’efficienza dei tradizionali schemi espliciti integrando le due classi di componenti con due passi temporali diversi, uno sottomultiplo dell’altro.
Günther, Kværno e Rentrop hanno introdotto una nuova analisi di questi schemi reinterpretando l’intero macropasso come uno schema Runge-Kutta Partizionato. Poiché l’efficienza di
uno schema multirate dipende in maniera essenziale da come viene realizzato l’accoppiamento
fra le due classi di varibili nel calcolo degli stadi, nelle applicazioni risulta importante avere
un criterio robusto per scegliere i coefficienti dell’accoppiamento quando si usino un numero
arbitrario di micropassi per le componenti veloci.
Nella prima parte, G. Visconti, descriverà una nuova classe di schemi Multirate RungeKutta espliciti nei quali l’accoppiamento tra le componenti lente e quelle attive è determinato
da una Estensione Continua Naturale dello schema Runge-Kutta di base. Gli schemi risultanti
sono molto versatili perché non dipendono da parametri liberi e quindi il numero di micropassi
può essere variato anche durante l’esecuzione. Verranno illustrate le condizioni d’ordine, un
metodo per l’analisi della stabilità lineare ed esempi fino al terzo ordine.
Nella seconda parte, M. Semplice, descriverà una estensione dei risultati precedenti al caso
dell’integrazione con “local timestepping” di leggi di conservazione su griglie non uniformi.
16.15-16.55 Metodi ghost-point alle differenze finite per PDE non lineari nella conservazione dei monumenti
A. Coco, Università di Catania
Il fenomeno della degradazione dei monumenti causata dalla solfatazione del marmo ha suscitato un sostanziale interesse nelle applicazioni. Tale fenomeno consiste nella trasformazione
di carbonato di calcio in gesso, a causa della presenza di biossido di zolfo nell’atmosfera, soprattutto in condizioni di umidit. Il modello che viene trattato nel presente seminario stato
descritto in [1] e diversi metodi numerici sono stati applicati, sia in 1D che in 2D per geometrie
semplici [2,3,4]. Uno dei lavori in corso quello di estendere tali metodi al caso di geometrie
complesse. In questo seminario viene presentato un possibile approccio, basato sulla descrizione
della geometria tramite level-set e imponendo le condizioni al bordo tramite una strategia di
tipo ghost-point, recentemente utilizzata con successo nell’ambito delle equazioni ellittiche [5].
Le equazioni del modello vengono discretizzate all’interno del dominio tramite differenze finite
e avanzate in tempo tramite Eulero Implicito. La risoluzione del sistema non lineare che ne
deriva effettuata tramite il metodo di Newton. Vengono quindi presentati i primi risultati
numerici in via preliminare.
Bibliografia
[1] D. Aregba Driollet, F. Diele, R. Natalini, A mathematical model for the SO2 aggression to
calcium carbonate stones: numerical approximation and asymptotic analysis, SIAM J. Appl.
Math. 64 (2004) 1636-1667.
[2] M. Donatelli, M. Semplice, S. Serra-Capizzano, Analysis of multigrid preconditioning for
implicit PDE solvers for degenerate parabolic equations, SIAM J. Matrix Anal. Appl. 32 (2011)
[3] M. Semplice, Preconditioned implicit solvers for nonlinear PDEs in monument conservation,
SIAM J. Sci. Comput. 32 (2010) 3071-3091.
[4] M. Donatelli, M. Semplice, S. Serra-Capizzano, AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation,
Applied Numerical Mathematics 68 (2013) 1-18.
[5] A. Coco and G. Russo. Finite-Difference Ghost-Point Multigrid Methods on Cartesian Grids
for Elliptic Problems in Arbitrary Domains. J. of Comp. Phys. , 241:464501, 2013.
16.55-17.35 A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts
M. Donatelli, Università dell’Insubria
Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang. The method in a nutshell consists of
a discrete Fourier transform-based alternating minimization algorithm with periodic boundary
conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this
talk, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous
alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution
problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an
accurate restoration of blurred and noisy images requires a proper treatment of the boundary.
A discrete version of our continuous alternating minimization algorithm is obtained following
two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur,
while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show
that our algorithm generates higher quality images in comparable running times with respect
to the Fast Total Variation deconvolution algorithm.
17.35-18.15 A sharp interface method for compressible multimaterial
H. Telib, Optimad Engineering
TBA
Tuesday
9.00-9.40 Schemi AP per problemi di controllo ottimo per sistemi iperbolici con
rilassamento
G. Albi, Università di Ferrara
Il seminario presentera alcuni recenti sviluppi su i metodi Runge–Kutta IMEX per la risoluzione
di problemi di controllo ottimo governati da un sistema di equazioni iperboliche. Discuteremo
nel dettaglio l’applicazione di questi schemi al modello di Goldstein–Taylor nel regime diffusivo,
ovvero nel limite dell’approssimazione iperbolica dell’equazione del calore, presentando alcuni
risultati numerici. Infine mostreremo alcune applicazioni di questi modelli nel campo dell’ingegneria biomedica.
Bibliografia:
[1] M. Herty, L. Pareschi and S. Steffensen, Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems, SIAM J. Num. Anal. to appear (arXiv:
1202.1166)
[2] G. Albi , M. Herty, C. Jrres and L. Pareschi, Asymptotic Preserving time-discretization of
optimal control problems for the Goldstein-Taylor model, preprint 2013
9.40-10.20 High-Order Semi-Implicit Schemes for time dependent partial differential equations
S. Boscarino, Università di Catania
In this paper we consider a new formulation of implicit-explicit methods for numerical discretization of time dependent partial differential equations. We introduce several semi-implicit
methods up to order three. These methods are particularly well suited for problems where the
stiff and the non-stiff components cannot be well separated. Finally, we conclude by a stability
analysis of the schemes for linear problems.
10.20-11.00 Automatic Domain Decomposition and Asymptotically Accurate schemes for the Vlasov-Poisson-BGK system with applications to plasmas
V. Rispoli, Università di Ferrara
We present an efficient strategy to face plasma physics simulations in which localized departures from thermodynamical equilibrium are present. The method relies on the introduction
of buffer zones which realize a smooth transition between the kinetic and the fluid regions.
We extend the techniques of dynamic coupling and of buffer zones, previously used for gas
dynamics, to the case of plasma simulations. The basic idea consists in using an hybrid scheme
in which both kinetic and fluid descriptions are considered and coupled together. Moreover,
we construct our scheme in order to solve the kinetic model by Asymptotic Preserving and
Asymptotically Accurate methods, in order to obtain high efficiency and to guarantee precision
of the proposed coupling strategy. The numerical scheme is validated and its performances are
analyzed by numerical simulations.
11.30-12.10 La modellazione del trasporto di soluti in canali meandriformi tramite
metodi DG ad alto ordine
V. Caleffi, Università di Ferrara
In questo lavoro viene presentato uno schema local discontinuous Galerkin (ldG), accurato
al terzo ordine, applicato ad un modello matematico bidimensionale, ottenuto per integrazione
su ciascuna verticale, che descrive lidrodinamica e il trasporto di contaminanti passivi in canali
meandriformi. La capacit del modello numerico proposto dagli autori di trattare correttamente
ed efficientemente domini computazionali con contorni curvilinei, preservando il corretto bilanciamento fra termine sorgente e divergenza del flusso in caso di quiete, di fondamentale
importanza per la corretta simulazione delle correnti a superficie libera in canalette di laboratorio e corsi dacqua naturali. Il modello matematico, originale in alcune sue parti, tratta gli
aspetti fisici pi significativi del deflusso della corrente in canali curvilinei (comprendendo lattrito al fondo, la dispersione della quantit di moto e dellinquinante e la diffusione turbolenta)
con un grado di completezza e complicazione omogeneo. Particolare attenzione rivolta alla
consistenza fra il modello matematico e le tecniche usate per lintegrazione numerica. Questo
aspetto particolarmente delicato in quanto si dimostra che la formulazione classica delle equazioni alle acque basse, comprensive dei termini diffusivi e dispersivi, non ottimale dal punto
di vista della sua integrazione numerica. Sono presentati due casi test per la validazione del
modello che consistono nel confronto fra risultati numerici e dati di laboratorio. Nel primo test
si considera la dispersione di un tracciante in condizioni stazionarie mentre nel secondo caso
viene investigata la propagazione di una nuvola di inquinante conseguente ad un rilascio istantaneo. I risultati dimostrano la potenzialit degli schemi ad alto ordine di accuratezza quando
applicati a problemi ingegneristici, inevitabilmente caratterizzati dalla complessit dei fenomeni
fisici coinvolti. Il buon accordo fra dati sperimentali e risultati numerici unulteriore conferma
della capacit dei modelli bidimensionali di cogliere gli aspetti fondamentali dei fenomeni di
convezione-diffusione senza la necessit di ricorrere a pi complesse modellazioni tridimensionali. Infine, interessante notare che la rappresentazione polinomiale della soluzione su ciascun
elemento di calcolo, tipica degli schemi discontinuous Galerkin, permette la valutazione della
curvatura delle linee di corrente, necessaria per la stima dei coefficienti di diffusione, in modo
semplice e consistente senza introdurre ricostruzioni arbitrarie.
14.30-15.10 Scattering matrix and WB scheme for non-relativistic limit of 1D Dirac
equations with potential
L. Gosse, IAC CNR Roma
The numerical approximation of one-dimensional relativistic Dirac wave equations is considered
within the recent framework consisting in deriving local scattering matrices at each interface of
the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously
shown that such a discretization preserves exactly the L2 norm despite being explicit in time.
This construction is well-suited for particles for which the reference velocity is of the order of
c, the speed of light. Moreover, when c diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let
a diffusive limit emerge numerically, like for discrete 2-velocity kinetic models. It is shown that
an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one.
15.10-15.50 Semi-Lagrangian schemes for the BGK model of the Boltzmann equation
G. Stracquadanio, Università di Parma
In this work we present high-order accuracy numerical methods for the BGK approximation
of the Boltzmann equation, based on the semi-Lagrangian feature, the stability property is
not restricted by the CFL condition. These aspects make them very attractive for practical
applications. Time integration is dealt with classical Runge-Kutta methods and also with BDF
methods, which are accurate and computationally less expensive. Numerical results and examples show that the schemes turn out to be reliable and ecient for the investigation of both
rareed and uid regimes of gasdynamics. Extensions and applications of such schemes to BGK
equations for gas mixtures are also presented. Joint work with G. Russo (Catania) and M.
Groppi (Parma).
Bibliografia:
[1] G.Russo, P. Santagati, S. B. Yun, Convergence of a semi-Lagrangian scheme for the BGK
model of the Boltzmann equation, SIAM J. Numer. Anal. 50 , no. 3, 11111135 (2012).
[2] P. Andries, K. Aoki, B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat.
Phys. 106, 9931018 (2002).
[3] E. Carlini, R. Ferretti, G. Russo, A weighted essentially nonoscillatory, large time-step
scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput 27, 10711091 (2005).
15.50-16.30 Numerical study of macroscopic pedestrian flow models
M. Twarogowska, Inria - Sophia Antipolis France
Growing population densities combined with easier transport lead to greater accumulation
of people and increase the risk of life threatening situations. Pedestrian traffic management is
aimed at designing walking facilities which follow optimal requirements regarding flow efficiency, pedestrian comfort and, above all, security and safety. From a mathematical point of view,
a description of human crowds is strongly non standard due to the intelligence and decisive
abilities of pedestrians. Their behaviour depends on the physical form of individuals and on
the purpose and conditions of their motion.
We consider two macroscopic models of pedestrian flow: the classical Hughes model, consisting of a scalar conservation law closed with a density-speed relation, and the second order
model given by the Euler equations for isentropic gas dynamics with source term. The desired direction of motion is determined by solving an eikonal equation with density dependent
running cost function standing for minimization of the travel time and avoidance of congested
areas.
We apply a mixed finite volume-finite element methods to solve the above problems on
unstructured meshes and present error analysis yielding a first order convergence. We compare
the models and show that the Hughes’ model is incapable of reproducing complex crowd dynamics. Finally, using the second order model, we study numerically the evacuation from a room
through a narrow exit. In particular, we analyze the accuracy of the approximation and the
influence of the model parameters and the geometry of the walking facility on the evacuation
time.
This research has been conducted in collaboration with Paola Goatin (INRIA Sophia Antipolis - Méditerranée, OPALE Project-Team) and Regis Duvigneau (INRIA Sophia Antipolis Méditerranée, OPALE Project-Team) 2
16.30-17.10 Discontinuous Galerkin Approximation of porous Fisher Kolmogorov
Equations
G. Naldi, Università di Milano
The reaction-diffusion equation plays an important role in dissipative dynamical systems for
physical, chemical and biological phenomena. For example, in chemical physics, in order to
describe concentration and temperature distributions, the heat and mass transfer are modeled
by a non linear diffusion term, while the rate of heat and mass production are described by
a non linear reaction term. These terms are often considered in the form of mass action law.
In population dynamics, where the focus is on the evolution of a population density, diffusion terms correspond to a random motion of individuals, and reaction terms describe their
reproduction and interaction, as in a predator-prey model.
Originally, continuum models of population spreading were based on linear diffusion. Then,
several authors have pointed out that these diffusion mechanisms can be more realistically
described by (degenerate) non linear diffusion models. The typical non linear reaction-diffusion
model is as follows:
∂u
− ∆p(u) = r(u)
∂t
∇p(u) · nΩ = 0
u|{t=0} = u0
in Ω × (0, +∞),
on ∂Ω × (0, +∞),
in Ω.
(1)
where Ω ⊂ Rd , with d = 1, 2, is a bounded domain, u is a density, p(u) is a suitable diffusivity,
r(u) a reaction function, and u0 the initial datum. If p(u) ' uγ , with γ > 0, equation (1) is
known as the porous media equation, which is degenerate in the sense that p0 (0) = 0. In this
case the solution can develop interfaces between the regions with zero and non zero population
density. These sharp fronted solutions represent population density profiles. The reaction is
modeled by a generalized Kolmogorov-Fisher term, namely r(u) = u(1 − uβ ).
In this talk, we consider the following generalized porous Fisher Kolomogorov model
∂u
D
−
∆uα+1 = Ru(1 − uβ )
∂t
α+1
∇uα+1 · nΩ = 0
u|{t=0} = u0
in Ω × (0, +∞),
on ∂Ω × (0, +∞),
in Ω,
(2)
with positive parameters R, β > 0, and α ≥ 1. From a numerical point of view, the discretization of this problem is challenging because the numerical scheme has to reproduce shock
waves or fronts of the analytical solutions, and preserve stability and invariance properties.
We present a new high-order numerical methods for the approximation of (2), based on a
discontinuous Galerkin space discretization and Runge-Kutta time stepping. These methods
are capable to reproduce the main properties of the analytical solutions. We present some
preliminary theoretical results and provide several numerical tests.