IperMiB2013 - peano.matapp.unimib.itpeano.matapp.unimib.it/ipermib2013/f/BookOfAbstracts.pdfIperMiB2013 15th Italian Meeting ... Dipartimento di Matematica e Applicazioni Università

IperMiB201315th Italian Meeting on Hyperbolic Equations

Book of Abstracts

September 11-13, 2013

Dipartimento di Matematica e ApplicazioniUniversità di Milano-Bicocca

2

Contents

Invited speakers . . . . . . . . . . . . . . . . . . . . . . . 5Francesco Caravenna . . . . . . . . . . . . . . . . . . 5Rinaldo M. Colombo . . . . . . . . . . . . . . . . . . 6Giacomo Dimarco . . . . . . . . . . . . . . . . . . . . 7Laurent Gosse . . . . . . . . . . . . . . . . . . . . . . 8Armando Majorana . . . . . . . . . . . . . . . . . . . 9Giuseppe Savare . . . . . . . . . . . . . . . . . . . . . 12

Contributed speakers . . . . . . . . . . . . . . . . . . . . 13Simonetta Abenda . . . . . . . . . . . . . . . . . . . 13Debora Amadori . . . . . . . . . . . . . . . . . . . . 14Paolo Antonelli . . . . . . . . . . . . . . . . . . . . . 15Davide Ascoli . . . . . . . . . . . . . . . . . . . . . . 16Sebastiano Boscarino . . . . . . . . . . . . . . . . . . 20Susanna Carcano . . . . . . . . . . . . . . . . . . . . 22Davide Catania . . . . . . . . . . . . . . . . . . . . . 23Fausto Cavalli . . . . . . . . . . . . . . . . . . . . . . 24Giuseppe Maria Coclite . . . . . . . . . . . . . . . . 26Armando Coco . . . . . . . . . . . . . . . . . . . . . 27Andrea Corli . . . . . . . . . . . . . . . . . . . . . . 29Marcello D’Abbicco . . . . . . . . . . . . . . . . . . . 32Edda Dal Santo . . . . . . . . . . . . . . . . . . . . . 34Maria Laura Delle Monache . . . . . . . . . . . . . . 35Lorenzo di Ruvo . . . . . . . . . . . . . . . . . . . . 37Carlotta Donadello . . . . . . . . . . . . . . . . . . . 38

4 CONTENTS

Natale Manganaro . . . . . . . . . . . . . . . . . . . 39Maxim V. Pavlov . . . . . . . . . . . . . . . . . . . . 40Livio Pizzocchero . . . . . . . . . . . . . . . . . . . . 41Vittorio Rispoli . . . . . . . . . . . . . . . . . . . . . 43Elena Rossi . . . . . . . . . . . . . . . . . . . . . . . 47Steave C. Selvaduray . . . . . . . . . . . . . . . . . . 48Laura V. Spinolo . . . . . . . . . . . . . . . . . . . . 50Marta Strani . . . . . . . . . . . . . . . . . . . . . . 51Paola Trebeschi . . . . . . . . . . . . . . . . . . . . . 52

Invited speakers 5

Invited speakers

Scaling limits and universality for random pinning models

Francesco Caravenna

The so-called ”random pinning model” was introduced in the chemistryand physics literature, to study the statistical behavior of a single poly-mer chain. From a mathematical viewpoint, it can be described as aprobability measure on a space of discrete paths: one takes a simplestochastic process (a Markov chain) and gives it a reward/penalty eachtime it visits a given site. When the rewards/penalties are chosen in arandom fashion, this model provides an example of a ”disordered sys-tem” which is amenable to a thorough mathematical investigation. Inthe large scale limit, many interesting features emerge, such as phasetransitions, that can be analyzed in detail. I will present a general re-view on the model, focusing on its universality properties in the weakcoupling regime, that lead to the construction of a continuum versionof the model.

6 Invited speakers

Nonlocal Balance Laws

Rinaldo M. Colombo

This presentation overviews recent results related to models based onnonlocal balance laws. The motivating applications are in vehicular traf-fic, crowd dynamics, individuals–populations interactions and predator–prey systems. We will address modeling issues, present analytical wellposedness theorems, state control problems and display the results ofnumerical integrations.

Invited speakers 7

On the time discretization of kinetic equations and relatedproblems

Giacomo Dimarco

In this talk we will discuss the time discretization of kinetic equationsof the Boltzmann type with particular focus on problems related tofluid dynamics and plasmas. The numerical solution of these modelshas encountered and still encounter many difficulties which are mainlydue to the stiffness of the equations close to the hydrodynamic or tothe diffusive regimes, when in other words, the mean free path goes tozero and to the discretization of the non linear multi dimension inte-gral which describes the interaction between the particles. In addition,the large dimensionality of the problems, the necessity to guaranteethe physical conservation properties as well as the positivity and theentropy inequality since they characterize the steady states makes theproblems very hard to solve in practice. In order to treat some of theabove problems, in this talk we discuss different time integration tech-niques which are particularly adapted to stiff kinetic equations. Weconsider both Implicit-Explicit Runge-Kutta methods as well as Expo-nential Runge-Kutta methods. We derive sufficient conditions in orderthat such methods are unconditionally asymptotically stable, asymp-totic and positivity preserving and asymptotically accurate. In the endwe will present several examples to show the behaviors of the schemesin different regimes.

8 Invited speakers

Residual distribution formalism and scattering matrices:application to a simple weakly nonlinear kinetic model of

population dynamics

Laurent Gosse

Residual distribution was introduced in an unpublished 1987 ICASE re-port by P.L. Roe and led to the development of many new schemes in 2D.Here we review its formulation and emphasize the links with the well-balanced ideas, especially the attention it draws onto the accurate com-putation of ”currents” (most of the numerical schemes instead focus onthe ”fields”). Restricting our attention to linear 1+1 kinetic equations,we show that an efficient manner of analyzing and correctly distributingresiduals is to employ a scattering matrix at each interface of the com-putational grid. Finally, we present a 2+2 Vlasov-Fokker-Planck kineticmodel of population dynamics within a (Strang-) dimensional-splittingframework. The dynamics are handled by a Morse-type self-consistentpotential, that we don’t want to compute by means of discrete convolu-tions (because of inevitable issues close to the computational domain’sborders). The aforementioned RD approach with S-matrices is set upand numerical results are presented for which the zero-divergence ofstationary macroscopic momentum is scrutinized.

Invited speakers 9

Macroscopic models derived from kinetic equations by usingthe discontinuous Galerkin method

Armando Majorana

We consider the kinetic equation

∂f

∂t+ a(ξ) · ∇xf + F(t,x) · ∇ξf = C(f) + S(t,x, ξ) ,

where the one-particle distribution function f depends on time t, po-sition x and velocity ξ. Here, a is a given vectorial function of thevelocity ξ, F is force term, which is given or related to another equation(for instance, the Poisson equation), C(f) is the linear or nonlinear col-lision operator, and S(t,x, ξ) is the source term. This model includesthe classical linear or nonlinear Boltzmann equation for rarefied gases,the Boltzmann equation for charge transport in semiconductors and theradiative transport equation.The heavy computational cost for solving kinetic equations, by usingsome deterministic schemes, explains why kinetic equations are tradi-tionally simulated by the Direct Simulation Monte Carlo (DSMC) meth-ods. The discontinuous Galerkin (DG) method seems very appropriatefor solving kinetic equations, since it guarantees a good accuracy witha reasonable computational cost.The DG scheme is a conservative scheme, that has the advantage offlexibility for arbitrarily unstructured meshes, and relies on an ade-quate choice of numerical fluxes, which handle effectively the interac-tions across element boundaries, to achieve stable and accurate algo-rithms for nonlinear hyperbolic conservation laws, nonlinear convectiondiffusion equations, etc.. The DG method allows to find approximatesolutions of the kinetic equation by solving a (usually large) set of or-dinary differential equations in time. Hence, the numerical distributionfunction f can be used to evaluate its moments.We propose a splitting of the numerical DG scheme, according to thefollowing remarks.

1. The variables x and ξ have a different physical meaning, and often,we are interested in the main moments of f instead of f itself;

10 Invited speakers

2. usually, we impose vanishing distribution function for large veloc-ities.

We suggest, as the first step, to discretize the kinetic equation withrespect only to the velocity variable ξ. Thus, we obtain a set of equa-tions, where the new unknowns depend only on the time t and thespatial coordinates x. These equations give macroscopic models, whichcan be studied analytically or numerically.

For instance, if the classical nonlinear Boltzmann equation is consid-ered, then a set of partial differential equations is obtained, where theunknowns Nαj(t,x) depend on the time and space coordinates. Theindex α labels the subset Cα of a partition of the velocity space andthe indexes j = 0, 1, .., 4 correspond to the five collision invariants. Theunknowns are related to the distribution function f(t,x, ξ), solution ofthe Boltzmann equation, since

Nα0(t,x) ≈∫

Cα

f(t,x, ξ) dξ ,

Nα1(t,x)Nα2(t,x)Nα3(t,x)

≈∫

Cα

f(t,x, ξ) ξ dξ ,

Nα4(t,x) ≈∫

Cα

f(t,x, ξ) |ξ|2 dξ .

The physical meaning of the new variables is evident; for instance,Nα0(t,x) is the density of particles at time t and position x havingvelocity belonging to the set Cα.The equations write

∂Nαj

∂t+

4∑

k=0

a(α)kj · ∇xNαk =

1

2

∑

β,γ

4∑

k,n=0

b(αβγ)jkn Nβk Nγn ,

where a(α)kj and b

(αβγ)jkn are the constant numerical parameters to be de-

termined. The new macroscopic model guarantees the conservation ofthe mass, momentum and energy.

Invited speakers 11

We will show some numerical examples for different kinetic equa-tions. In these cases the deterministic DG method furnishes an accuratedescription of the gas dynamics.

12 Invited speakers

Lagrangian solutions to the one-dimensional Euler-Poissonsystem

Giuseppe Savare

We present a Lagrangian formulation of the pressureless Euler-Poissonsystem in one dimension, as a particular case of a sticky particle modeldriven by a (non-smooth) interaction potential. Tools from differentialinclusions and optimal transport can be applied to provide existence ofa solution; uniqueness and an explicit representation formula can alsobe proved in the attractive case. (In collaboration with Y. Brenier, W.Gangbo, M. Westdickenberg)

Contributed speakers 13

Contributed speakers

Grassmannians and multi-soliton solutions to KP-II: analgebraic geometric approach

Simonetta Abenda

The problem of characterizing classes of solutions to the KP-II equationhas attracted the attentions of many outstanding mathematicians sinceits introduction. A (N,M)-soliton solution to KP-II is a real boundedregular solution u(x; y; t) which has M line soliton solutions in asymp-totics in the x, y plane whose directions are invariant w.r.t. to t. Thesesolutions are defined as a torus orbit on the Grassmannian manifoldGr(N,M). They may be classified in terms of the matroid strata in thetotally non-negative part of Gr(N,M) and to each such point there isassociated a real and totally non-negative matrix A in reduced echelonform. In a recent paper, Yu. Kodama and L. Williams associate thedirections of the asymptotic solitons to the combinatorial classificationof the Grassmannian.

In this talk we address the classification problem of such multi-solitonic solutions from another point of view: we associate to a genericpoint in the totally non negative part of Gr(N,M) a compatible setof (N + 1) divisors sitting on an m-curve (perturbation of the ratio-nal curve associated to the multi-solitonic solution) and give an explicitrepresentation of the matrix A in terms of such system of divisors. Thechoice of such divisors is unique under the requirement that the matrixA is irreducible and in row echelon form.The results have been obtained in collaboration with Petr G. Grinevich

(Landau institute of Physics).

14 Contributed speakers

Error estimates for well-balanced Godunov schemes on scalarbalance laws

Debora Amadori

The talk will concern the approximation of conservation laws with sourceterm:

∂tu+ ∂xf(u) = k(x)g(u) .

We aim at comparing two different approaches: the Time-Splitting (TS)and Well-Balanced (WB) approach. We focus on the case of ”accretivesources”, in the sense that supx,u k(x)g′(u) > 0, and assume a non-resonance condition, f ′ > 0.

Error estimates for Time-Splitting approximations were proved inthe literature and read as:

‖u∆t(t, ·)− u(t, ·)‖L1(R) ≤ C exp(sup[k(x)g′(u)] t)√∆t , ∆t > 0 .

In these proofs, the Gronwall Lemma is applied and an exponentialincrease in time may occur when the source term is accretive. Numericalproofs show that this exponential amplification can actually occur.

We will provide rigorous L1 error estimates for a WB approximationof Godunov type, showing that for this class of approximation the errorgrows at most only linearly in time.

This is a joint work ([1]) with Laurent Gosse, IAC-CNR, Rome.

[1] D. Amadori, L. Gosse, Transient L1 error estimates for well-balanced

schemes on non-resonant scalar balance laws, J. Differential Equa-tions, 255 (2013), 469–502.


On nonlinear Schrodinger type equations with nonlineardamping

Paolo Antonelli

In this talk I report about some recent results obtained in collabora-tion with Remi Carles (Montpellier) and Christof Sparber (Chicago).We analyse the effect of nonlinear damping in augmented nonlinearSchrodinger equations: we show it prevents finite time blow-up in severalsituations. Furthermore, in the presence of a quadratic confining poten-tial, the solution is shown to asymptotically converge to zero, whereasin the case without external potential it is proven that the solution doesnot go to zero in general, due to non-trivial scattering. Such modelsarise in various areas of physics, such as non-equilibrium Bose-Einsteincondensates.


Wellposedness in the Lipschitz class for a quasi-linearhyperbolic system arising from a model of the atmosphere

including water phase transitions

Davide Ascoli

S. C. Selvaduray and H. Fujita Yashima introduced in [SF] a model forair motion in three-dimensional space, including water phase transitionsfrom one to any other of its three states: gaseous, liquid and solid;this model was the complete version of the simplified one discussed in[BAF], and it represents a sufficiently detailed mathematical descriptionof the physical phenomena involved. In [SF] an existence and uniquenesstheorem was proved for a modified problem in which vapor density isreplaced by a local average of it.

Now, in [AS], instead, we make a first step in the direct study -without smoothing the vapor density - of the model discussed in [SF]:we suppose that air velocity v(t, x) and temperature T (t, x) are known,and prove wellposedness for the hyperbolic part of the system introducedin [SF], which is of the kind:

(Q)

∂tρ+∇x · (v ρ) = 0,

∂tπ +∇x · (v π) = P(π, σ, ν),

∂tσ +∇x · (uσ) + ∂m[sl (π − πl(T ))σ] = S(π, σ, ν),∂tν +∇x · (w ν) + ∂m[ss (π − πs(T )) ν] = N (π, σ, ν).

Our unknown functions are:ρ(t, x): the density of dry air,π(t, x): the density of water vapor,σ(t, x,m) represents, at time t, the total mass, per unit volume cen-

tered at x, of the water contained in droplets of mass m,ν(t, x,m): analogously, the density of water at the solid state, also

dependent on the mass m of ice particles.For u and w, the velocities of droplets and of ice particles, we suppose

a dependence on t, x and m; this one may well include some link withv; sl(m) and ss(m) are proportional to the surfaces of droplets andice particles respectively of mass m, but since we suppose that they


always contain an aerosol kernel and since it does not make sense toconsider too big drops or ice particles, we assume sl(m) = ss(m) = 0for m ≤ ma > 0 and that sl and ss are bounded. πl and πs are thesaturated vapor densities relative to the liquid and to the solid states.

System (Q) is a diagonal quasilinear hyperbolic one; the semilinearterms S andN contain both quadratic collisional integrals and Lipschitzcontinuous terms; these last ones prevent from applying Kato’s results of[K] – that concern also systems with terms in which dependence on thesolution is not pointwise – in order to obtain classical solutions (we evenshow that there exist non-differentiable solutions). Wellposedness inthe Lipschitz class for diagonal quasilinear systems with Lipschitz con-tinuous bounded coefficients and inhomogeneous terms (not includingintegral terms) in the whole space was proved by M. Cinquini Cibrario;see also [J] for a semigroup approach.

For system (Q) we prove local in time wellposedness for boundedlysupported solutions whose initial data are Lipschitz continuous func-tions of x and m, where x ∈ Ω – a bounded open set of R3 – andm∈ ]0,+∞[; velocities are supposed to be tangent to the strongly Lip-schitz continuous boundary of Ω. The solutions (ρ, π, σ, ν) lie in

L∞(]0, τ∗[;W 1,∞(Ω))2 × L∞(]0, τ∗[;W 1,∞(Ω+))2,

where Ω+ = Ω×]0,+∞[, and τ∗ is sufficiently small, and they dependlocally Lipschitz-continuously, in the L∞-norm, on initial data, temper-ature and velocities; they lie also in

W 1,q(]0, τ∗[;L∞(Ω))2 ×W 1,q(]0, τ∗[;L∞(Ω+))2,

where q ∈ [1,∞] depends on a time regularity of velocities and temper-ature.

In order to prove our theorem, we first consider a linearization of thesystem such that its solutions are non-negative; it is a linear diagonalsystem, whose coefficients are Lipschitz continuous in x and m. Sinceour problem is considered on a bounded open set and we also need es-timates, we are led to prove preliminary results. The main proof we


perform is based on L∞-estimates as limits of Lp-estimates (this proce-dure was already used by E. Tadmor), extension to R3, regularizationof the coefficients and of the inhomogeneous terms, and converging sub-sequences. A direct use of integration along characteristics is avoided.

In order to find the solution of the quasi-linear problem, we thenapply the contraction theorem in a closed subset of

L∞([0, τ∗];W 1,∞(Ω))2 × L∞([0, τ∗];W 1,∞(Ω+))2.

The contraction is with respect to the L∞-norm.We need, therefore, suitable majorizations of the L∞-norms of the

solutions of the linearized problem and of their derivatives with respectto x and to m.

Last, we prove continuous dependence on data, temperature andvelocities.

This is a joint work with Steave C. Selvaduray.

References

[AS] Ascoli, D., Selvaduray, S.C., Wellposedness in the Lipschitz classfor a quasi-linear hyperbolic system arising from a model of theatmosphere including water phase transitions, Quad. Scient. Dip.

Matem. Univ. Torino, 6, 2012, 22 pp.,https://dspace-unito.cilea.it/handle/2318/824.

[BAF] Belhireche, H., Aissaoui, M.Z., Fujita Yashima, H., Equationsmonodimensionnelles du mouvement de l’air avec la transition dephase de l’eau, to appear on Sciences et Technologie, Univ. Con-

stantine.

[J] F. Jochmann, A semigroup approach to W 1,∞-solutions to a classof quasilinear hyperbolic systems, J. Math. Anal. Appl. 187,723-742, 1994.

[K] T. Kato, The Cauchy problem for quasi-linear symmetric hyper-bolic systems, Arch. Rat. Mech. Anal. 58, 181-205, 1975.


[SF] Selvaduray, S., Fujita Yashima H., Equazioni del moto dell’ariacon la transizione di fase dell’acqua nei tre stati: gassoso, liquidoe solido, Atti dell’Accademia delle Scienze Torino, 2011.


High–Order Asymptotic–Preserving Methods for nonlinearhyperbolic to parabolic relaxation problems

Sebastiano Boscarino

We consider nonlinear hyperbolic systems with stiff nonlinear relax-ation. In the parabolic scaling, when the stiffness parameter ε → 0, thedynamics is asymptotically governed by effective systems which are ofparabolic type and may contain possibly degenerate and fully nonlineardiffusion terms. In [3] we study and analyze several model problems withlinear and nonlinear relaxation, including Kawashima-LeFloch’s model[4]. In order to solve numerically such problem we use a method of line.First we discretize in space by conservative finite difference methods inthe sense of Shu and obtain a large system of ODE’s. For such sys-tem, we introduce implicit–explicit (IMEX) methods which are basedon Runge–Kutta (R–K) discretization in time. We shall show that theuse of classical IMEX R-K methods will lead to consistent explicit dis-cretization of the limit equation, i.e. ε → 0. Because of this, the limitscheme will suffer of the typical parabolic CFL restriction ∆t ∝ ∆x2.Furthermore in some cases such restriction does not allow the computa-tion of the solution due to the unfoundedness of the diffusion coefficient(super-linear case). To overcome such drawback one can make use of apenalization technique, based on adding two opposite terms (see for de-tails [1, 2]). By this technique in the limit ǫ → 0 the IMEX R-K schemeconverges to an implicit or semi-implicit method, then this IMEX R-Kschemes are able to solve the hyperbolic system containing nonlinear re-laxation without any restriction on the time step. In [3] a very efficientIMEX R-K scheme has been introduced, in order to solve nonlinear dif-fusion equation. The method we propose is stable, linearly implicit andcan be designed up to any order of accuracy. Several applications tosome test problems will be presented in order to show the effectivenessof the new approach.

This is a joint work with Philippe G LeFloch and Giovanni Russo.

[1] S. Boscarino, L. Pareschi, G. Russo, Implicit-Explicit Runge-Kutta

schemes for hyperbolic systems and kinetic equations in the diffusion


limit, SIAM J. SCI. COMPUT., Vol. 35, No. 1, pp. A22–A51, 2013.

[2] S. Boscarino, G. Russo, Flux-Explicit ImEx Runge-Rutta Schemes

for Hyperbolic to Parabolic Relaxation Problems, SIAM J. Numer.

Anal., SIAM J. NUMER. ANAL., Vol. 51, No. 1, pp. 163–190,2013.

[3] S. Boscarino, P. G. LeFloch, and G. Russo, High-order asymptotic–

preserving methods for fully nonlinear relaxation problems. submit-ted to SIAM J. Sci. Comput., Preprint: arxiv.org/pdf/1210.4761.

[4] S. Kawashima, P. G. LeFloch, in preparation.


A Discontinuous Galerkin Method for the Simulation ofMultiphase Flows

Susanna Carcano

The mixture of gas and solid particles in non-equilibrium conditionsthat compose a multiphase flows is described with a set of coupled par-tial differential equations for the mass, momentum and energy of eachspecies. Solid particles and the gas phase are considered as interpene-trating continua, following an Eulerian-Eulerian approach. Each speciesis compressible and inviscid. The gas and particles dynamics are cou-pled through the drag term in the momentum equations and the heatexchange term in the energy equations.Following the methods of lines, a p-adaptive Discontinuous Galerkinspace discretization is introduced first, then different explicit time dis-cretization schemes are adopted. A monotonization technique is intro-duced on the advective terms of the system. An automatic criterion isintroduced to adapt the local number of degrees of freedom and to im-prove the accuracy locally. The technique is improved in order to takeinto account for the presence of discontinuities in the solution. Theemployed technique is simple and relies on the use of orthogonal hier-archical tensor-product basis functions.The numerical model is applied and tested to several relevant 1D testcases, in order to assess its accuracy and stability properties. Compar-isons between different time discretization schemes and monotonizationapproaches are presented. Moreover we analyze the efficiency of thep-adaptivity approach.


Analysis of the linearized MHD-Maxwell interface problem

Davide Catania

We consider the free boundary problem for the plasma-vacuum interfacein ideal compressible magnetohydrodynamics (MHD). In the plasma re-gion, the flow is governed by the usual compressible MHD equations,while in the vacuum region we consider the Maxwell system for theelectric and the magnetic fields. We assume that the free interface isa tangential discontinuity: the total pressure is continuous, the mag-netic field is tangent to the boundary and the electric and the magneticfields are related through a suitable physical condition. The plasmadensity does not go to zero continuously at the interface, but has ajump, meaning that it is bounded away from zero in the plasma regionand it is identically zero in the vacuum region. This is a mathematicalmodel for plasma confinement by magnetic fields, such as in tokamaks,or for the motions of stellar coronas, when magnetic fields are taken intoaccount.

Under suitable stability conditions satisfied at each point of theplasma-vacuum interface, we prove an a priori estimate for the solu-tion to the linearized problem in (conormal) Sobolev spaces. This isa preliminary step in order to prove well-posedness for the nonlinearproblem, which we aim to handle in a forthcoming paper.

This is a joint work with Marcello D’Abbicco and Paolo Secchi.


Numerical approximation of multiscale hyperbolic systems

Fausto Cavalli

In this talk we compare and study several numerical approaches to ob-tain all-speed, asymptotic preserving and unconditionally stable numer-ical schemes for hyperbolic systems containing stiff relaxation sourceterms. Such models arise in many physical problems, as in the modellingof multiphase flows involving phase transitions, kinetic-type phenomena,semiconductor devices. As an example, we can mention isentropic Eu-ler equations, Euler equations with linear friction, radiative transfertmodels, Euler-Poisson systems [6,7]. The presence of the stiff relax-ation term allows to describe distinct physical time-scales terms, whichcan exhibit different behaviours, from hyperbolic to diffusive regime.The study and the numerical approximation of these models is particu-larly challenging, as several aspects have to be handled. First of all, wewant that the numerical scheme be accurate and stable for all the timesscales, in particular when, for small values of the relaxation parame-ter, the characteristic speeds become very large and this would affectthe accuracy increase the computational costs and [1-3]. Moreover, thenumerical scheme has to be asymptotic preserving, i.e. to be able toreproduce the right equilibrium for vanishing values of the relaxationparameters, also when the initial data are far from the equilibrium [4].Finally, we want to avoid a dependence ∆t = O(ǫh) of the time integra-tion step with respect to the relaxation parameter ǫ and the mesh sizeh, which would be too restrictive when ǫ ≪ 1, as well as the parabolicstiffness ∆t = O(h2) of explicit schemes for diffusive equations. Wepresent several numerical experiments and comparisons together withsome preliminary theoretical results and considerations.

Joint work with G. Naldi.

[1] Jin S, Pareschi L, Toscani G, (2001) Uniformly accurate diffusiverelaxation scheme for multiscale transport equations, SIAM Journalon Numerical Analysis; 38(3):913-36.

[2] Haack,J. and Jin,S. and Liu,J., (2012), An all-speed asymptotic-


preserving method for the isentropic Euler and Navier-Stokes equa-tions. Communications in Computational Physics 12(4) 955-980.

[3] Cordier F, Degond P, Kumbaro A. (2012)An asymptotic-preserving all-speed scheme for the euler and navier-stokes equations. Journal of Computational Physics;231(17):5685-704.

[4] Filbet F, Jin S. A class of asymptotic-preserving schemes for ki-netic equations and related problems with stiff sources. Journal ofComputational Physics 2010;229(20):7625-48.

[5] Boscarino, S., Pareschi, L., Russo, G. (2013). Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in thediffusion limit. SIAM Journal on Scientific Computing, 35(1), A22-A51.

[6] Boscarino, S., Russo, G. (2013). Flux-explicit IMEX Runge-Kuttaschemes for hyperbolic to parabolic relaxation problems. SIAMJournal on Numerical Analysis, 51(1), 163-190.

[7] Cavalli, F., Naldi, G., Puppo, G., Semplice, M. (2007). High-order relaxation schemes for nonlinear degenerate diffusion prob-lems. SIAM Journal on Numerical Analysis, 45(5), 2098-2119.


Systems with moving boundaries

Giuseppe Maria Coclite

We consider a system of scalar balance laws in one space dimensioncoupled with a system of ordinary differential equations. The couplingacts through the (moving) boundary condition of the balance laws andthe vector fields of the ordinary differential equations. We prove theexistence of solutions for such systems passing to the limit in a vanishingviscosity approximation. The results were obtained in collaborationwith Mauro Garavello (Milano-Bicocca).


Third order shock capturing schemes on non graded adaptivemeshes in two space dimensions

Armando Coco

In this talk we introduce and study a third order accurate scheme forhyperbolic systems of conservation laws in two-dimensional spaces usingcompact CWENO schemes on an adaptive grid. An error indicatorbased on numerical entropy production gives information on the localquality of the numerical solution in order to decide where the grid shouldbe locally refined or coarsened. For time integration, we will restrictourselves to classic explicit TVD-Runge-Kutta schemes as the focus ofthe present talk is on the CWENO reconstruction. In order to constructa third order scheme suitable for an adaptive mesh refinement setting,we extend the idea of the CWENO reconstruction [1] to nonuniformmeshes. In one space dimension the extension is straightforward, whilein higher dimensions a complication arises due to the variable cardinalityof the set of neighbors of a given cell. This issue is overcome by usingan interpolation procedure of the cell averages of the neighbor cells ina Least-Square sense. We show that the accuracy of the reconstructionis the same that one would expect in the interpolatory case. Severalnumerical tests in one and two space dimensions are provided in order todemonstrate both the properties of the reconstruction and corroboratethe power of adaptive schemes in this kind of problems. A simple scalingargument shows that third order schemes for piecewise smooth solutionsare more effective than second order ones for one space dimension, whilethey are not so effective in three space dimensions. The effectiveness ofthe approach for two space dimensions is validated numerically on theseveral tests performed.

[1] D. Levy, G. Puppo, and G. Russo. Compact central WENO schemesfor multidimensional conservation laws. SIAM J. Sci. Comput.,22(2):656–672, 2000.

[2] G. Puppo and M. Semplice. Numerical entropy and adaptivity forfinite volume schemes. Commun. Comput. Phys., 10(5):1132–1160,


2011.

[3] D. Levin. The approximation power of moving least-squares. Math.Comp., 67 (224):1517–1531, 1998.


Parabolic Approximations of Diffusive-Dispersive Equations

Andrea Corli

We are interested in the following initial-value problem for a diffusive-dispersive conservation law:

uεt + f(uε)x = εuε

xx + γε2uεxxx in ΩT : = R× (0, T ), T > 0,

uε(., 0) = u0 in R.(1)

Here, f : R → R denotes a smooth flux with f(0) = 0, u0 : R → R isthe initial datum and ǫ, γ are positive real parameters.

System (1) is a simple model to describe phase transition dynam-ics when viscous and capillary effects are taken into account. To seethis, consider the sharp interface limit ε → 0, where (1) reduces to thehyperbolic conservation law

ut + f(u)x = 0. (2)

Assume that the flux f has an inflection point, which occurs in phasetransitions. In such a case, shock-wave solutions to (2) that connectend states u−, u+ in different regions of convexity may be interpretedas phase boundaries; in particular, such waves can be undercompressive.The crucial observation is that these undercompressive shock waves pos-sess a diffusive-dispersive profile. Moreover, solutions of (1) convergefor ε → 0 to a weak solution of the corresponding initial-value problemfor (2).

Coming back to (1), it is obvious that both its analytical and numer-ical treatment gets intricate due to the appearance of the third-orderterm. Therefore we introduce lower-order approximations of (1) thatkeep the property of allowing traveling-wave solutions associated withundercompressive waves of (2) and reduce to (2) in the sharp-interfacelimit ε → 0.

One possibility for lower-order approximations for (1) is the nonlocalequation

uε,αt + f(uε,α)x = εuε,α

xx + γα (Kε,α ∗ uε,α − uε,α)x , (3)


which depends on the coupling parameter α > 1. For fixed ε > 0, formalasymptotics suggests that solutions uε,α of (3) converge for α → ∞ tosolutions of (1). Though this has not been proven so far for arbitrarykernels Kε,α it holds true indeed for the choice [1]

Kε,α(x) =

√α

2εe−

√

α

ε|x|.

In this case (3) is equivalently rewritten in the local parabolic-ellipticform


xx − α(uε,α − λε,α)x,

−γε2λε,αxx = α(uε,α − λε,α),

in ΩT . (4)

Instead of purely static equations as (4)2, one may also think to se-lect time dependent operators. The simplest choice in this context andindeed the main topic of [2] is


xx − α(uε,α − λε,α)x,

βλε,αt − γε2λε,α

xx = α(uε,α − λε,α),in ΩT , (5)

for 0 < β ≤ 1. By (5)2 we can guess that λε,α and uε,α converge to thesame limit as α → ∞. If we plug (5)2 into (5)1, a necessary conditionto recover (1)1 in the limit α → ∞ is that

β = β(α) → 0 for α → ∞. (6)

About the initial conditions for (5) we simply require

uε,α(·, 0) = λε,α(·, 0) = u0 in R. (7)

As mentioned above, we are particularly interested in flux functions fthat are neither concave nor convex. To avoid technicalities, we focus ontwo classes of flux functions: we assume that the flux function f : R → R

has bounded derivatives or it coincides with the nonconvex flux

f(u) = u3. (8)


The results we obtain are the following. First, we prove the existenceof traveling waves to (5) for the cubic flux (8), for particular values ofγ. The proof relies on geometric singular perturbation theory.

Second, we prove that a (sub)sequence of solutions to (5)-(7) con-verges to a weak solution of the diffusive-dispersive problem (1) forα → ∞. The scaling of β with respect to the coupling parameter α iscrucial: we sharpen 6 to

β = β(α) = O(α−1) for α → ∞.

We also obtain an explicit convergence rate with respect to the couplingparameter.

Third, we study the sharp interface limit ε → 0, now using for fixedα the scaling

β = β(ε, α) = εO(α−1) for ε → 0.

We prove by the compensated compactness approach in Lp spaces thatsolutions of (5)-(7) converges to a weak solution of (1) with initial datumu0.

Joint work with Christian Rohde and Veronika Schleper

[1] A. Corli and C. Rohde. Singular limits for a parabolic-elliptic reg-ularization of scalar conservation laws. J. Differential Equations,253(5):1399–1421, 2012.

[2] A. Corli, C. Rohde and V. Schleper. Parabolic approximations ofdiffusive-dispersive equations. Preprint, 2013.


Semilinear wave equation with time-dependent damping

Marcello D’Abbicco

In this talk we discuss the application of some linear decay estimates toderive the global existence of the small data solution to

utt −u+ b(t)ut = f(t, u) , t ≥ 0 , x ∈ Rn ,

u(0, x) = u0(x) ,

ut(0, x) = u1(x) .

(9)

Here |f(t, u)| . (1+ t)γ |u|p and data are small in (L1 ∩H1)× (L1 ∩L2)or in some weighted energy space H1(ρ)× L2(ρ).

We assume that the damping term b(t) > 0 is effective, that is, theL1−L2 decay rate for (9) is the same of the corresponding heat equationb(t)ut − u = 0, and that b(t) is sufficiently smooth. We may provethat for p > p(n, γ) the solution to (9) is global and it satisfies decayestimates with the same decay rate of the linear ones [3]. We also provethat in some cases the exponent p is critical, i.e. no global solutionexists to (9) if p ≤ p, for suitable, arbitrarily small data [2]. For γ = 0the critical exponent is p = 1+2/n, the same one found in the constantcoefficient case [5,6]. This exponent goes back to Fujita’s paper [4] aboutglobal existence for the semilinear heat equation ut −u = up.

A special case of damping coefficient for (9) is b(t) = µ(1 + t)−1: inthis case, Fujita exponent is obtained for sufficiently large µ, whereasfor small µ the damping becomes non effective [1].

[1] M. D’Abbicco The Threshold between Effective and Noneffective

Damping for Semilinear Waves, arXiv:1211.0731 [math.AP]

[2] M. D’Abbicco, S. Lucente, A modified test function method for

damped wave equations, Adv. Nonlinear Studies, to appear,arXiv:1211.0453 [math.AP].

[3] M. D’Abbicco, S. Lucente, M. Reissig, Semilinear wave equations

with effective damping, Chinese Ann. Math. 34B (2013), 3, 345–380, doi:10.1007/s11401-013-0773-0; arXiv:1210.3493 [math.AP]


[4] H. Fujita, On the blowing up of solutions of the Cauchy Problem for

ut = u+ u1+α, J. Fac.Sci. Univ. Tokyo 13 (1966), 109–124.

[5] R. Ikehata, K. Tanizawa, Global existence of solutions for semilinear

damped wave equations in RN with noncompactly supported initial

data, Nonlinear Analysis 61 (2005), 1189–1208.

[6] G. Todorova, B. Yordanov, Critical Exponent for a Nonlinear Wave

Equation with Damping, Journal of Differential Equations 174 (2001),464–489.


On the front tracking for 2× 2 systems of conservation laws

Edda Dal Santo

We deal with a simplified version of the classical front-tracking algorithmfor a general genuinely nonlinear/linearly degenerate 2 × 2 system ofconservation laws. Besides, we address the case of the system

ht − (hp)x = (p− 1)h,pt + ((p− 1)h)x = 0,

arising in the study of granular flows in [1]. For the associated homoge-neous part we prove the well-definiteness of a front-tracking algorithmand its convergence to a weak entropic solution, in the case of possiblylarge BV initial data.

[1] D. Amadori and W. Shen, Global Existence of Large BV Solutions

in a Model of Granular Flow, Comm. Partial Differential Equations34 (2009), 1003–1040.


Traffic flow optimization on networks

Maria Laura Delle Monache

The aim of this work is to introduce a model for optimizing trafficflow on roundabouts. We refer to the model described in [3] and weapply it to roundabouts. Roundabouts can be seen as particular roadnetworks and they can be modeled as a concatenation of junctions. Eachjunction is described by a coupled PDE-ODE system. We focus on aroundabout with three roads that can be modeled as a 2x2 junctionwith two incoming and two outgoing roads. In particular, each junctionhas one incoming mainline, one outgoing mainline and a third link withoutgoing and incoming fluxes. The third road is modeled with a bufferof infinite capacity for the incoming flux and with an infinite sink for theoutgoing one. The mainline evolution is described by a scalar hyperbolicconservation law, whereas the buffer dynamics is described with an ODEwhich depends on the difference between incoming and outgoing fluxeson the link. At each junction the Riemann problem is uniquely solvedusing a right of way parameter and solutions are constructed via wave-front tracking. We aim at minimizing the total travel time of cars on thenetwork. Such functional is optimized for a simple network consistingof a single 2x2 junction. The optimization is made with respect to theright of way parameter, obtaining an analytical solution. Then, throughsimulations, the traffic behavior for more complex networks is studied.Compared to existing models (see [1, 2]), our model based on the PDE-ODE coupled system is able to describe different dynamics at junctions,which were not included in the previous ones.Joint work with Legesse Lemecha Obsu, Paola Goatin and Semu MitikuKasa.

References

[1] A. Cascone, C. D’Apice, B. Piccoli and L. Rarita Optimization oftraffic on road networks, Mathematicals Models and Methods inApplied Sciences, 17(10) (2007), 1587–1617.


[2] A. Cutolo, C. D Apice and R. Manzo, Traffic optimization at junc-tions to improve vehicular flows, ISRN Applied Mathematics, (2011),19pp.

[3] M. L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P.Goatin and A. M. Bayen, A PDE-ODE model for a junction withramp buffer, preprint, http://hal.inria.fr/hal-00786002.


Convergence of the Ostrovsky Equation to theOstrovsky–Hunter One

Lorenzo di Ruvo

The non-linear evolution equation

∂x(∂tu+ u∂xu− β∂3xxxu) = γu, (10)

with β, γ ∈ R was derived by Ostrovsky to model small-amplitude longwaves in a rotating fluid of finite depth. This equation generalizes theKorteweg-deVries equation (that corresponds to γ = 0) by an addi-tional term induced by the Coriolis force. It is deduced considering twoasymptotic expansions of the shallow water equations, first with respectto the rotation frequency and then with respect to the amplitude of thewaves.If we sent β → 0 in (10), then we pass from (10) to the equation

∂x(∂tu+ u∂xu) = γu, (11)

known as Ostrovsky–Hunter equation.We study the dispersion-diffusion limit for (11). Therefore, we considerthe following third order approximation

∂tuε,β + uε,β∂xuε,β − β∂3xxxuε,β = γPε,β + ε∂2

xxuε,β , t > 0, x ∈ R,

∂xPε,β = uε,β , t > 0, x ∈ R.

Using the compansated compactness, we prove that there exists uβn,εn ,subsequence of uβ,ε, such that uβn,εn → u, where u is a distributionalsolution of (11).Moreover, choosing β in suitable way, we prove that there exists uβn,εn ,subsequence of uβ,ε, such that uβn,εn → u, where u is the unique entropysolution of (11).

Joint work with: G. M. Coclite (Department of Mathematics University of

Bari via E. Orabona 4, 70125 Bari, Italy), K. H. Karlsen (Centre of Mathe-

matics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern,

N–0316 Oslo, Norway).


Crowd dynamics and conservation laws with nonlocalconstraints

Carlotta Donadello

In this talk we propose a macroscopic model for pedestrians evacuatinga narrow corridor through a single exit. By coupling the 1-D hyper-bolic equation describing the conservation of the density of pedestrianswith a non–local constraint which depends on the density itself, ourmodel tries to capture the gradual fall in the efficiency of the exit as thepedestrians clog it. We show that the solutions of the Riemann prob-lem with piecewise constant non–local constraint are not unique, norself–similar in general. Then we obtain results on the well–posednessof the Cauchy problem with Lipschitz non–local constraint by a proce-dure that combines the wave–front tracking algorithm with the operatorsplitting method. This work is part of an ongoing research project incollaboration with M.D. Rosini and B. Andreianov.


Riemann problems for a traffic flow model

Natale Manganaro

In view of solving in a closed form initial and/or boundary value prob-lems of interest in nonlinear hyperbolic and dissipative wave processesit is considered a reduction approach based upon appending differentialconstraints to quasilinear nonhomogeneous hyperbolic systems of firstorder PDEs [1], [2]. In this context a nonhomogeneous first order hy-perbolic system describing traffic flow based on the Aw-Rascle model [3]is analyzed thoroughly and a classification of possible constraints alongwith sets of consistent response functions involved therein is worked outwhereupon the classes of corresponding exact solutions are determined.To some extent these solutions generalize the classical simple wave so-lutions and may also incorporate dissipative effects. Furthermore, inorder to solve Riemann problems and generalized Riemann problems,rarefaction wave-like solutions are determined. Finally applications tosome traffic flow problems are given.

References

[1] S. V. Meleshko, One class of solutions for the systems of quasilineardifferential equations with many independent variables, Chisl.metodymech.splosh.sredy, 12 (1981) 87-99.

[2] N. Manganaro and S. Meleshko, Reduction procedure and general-ized simple waves for systems written in Riemann variables, Nonlin-ear Dynamics, 30 (2002) 87-96.

[3] A. Aw and M. Rascle, ”Resurrection of second order models of trafficflow, SIAM J. Appl. Math., 60, 916-938 (2000).


Integrability of Russo-Smereka kinetic equation

Maxim V. Pavlov

In this talk we consider the kinetic equation derived by G. Russo and P.Smereka for description of motion of compressible bubbles in a perfectfluid (see [3]). Later this kinetic equation was also investigated in fewpapers by A. Chesnokov, V. Teshukov and S. Gavrilyuk (see [1], [4]).

We prove that this kinetic equation is an integrable equation by themethod of hydrodynamic reductions (see detail in [2]).

We construct some families of solutions in implicit form by the Gen-eralized Hodograph Method (see detail in [5]).

References

[1] S. Gavrilyuck and V. Teshukhov, Kinetic model for the motion ofcompressible bubbles in a perfect fluid. Eur. J. Mech. B/Fluids21(2002) 469–491.

[2] J. Gibbons, S.P. Tsarev, Reductions of the Benney equations, Phys.Lett. A 211 No. 1 (1996) 19-24. J. Gibbons, S.P. Tsarev, Conformalmaps and reductions of the Benney equations, Phys. Lett. A 258No. 4-6 (1999) 263-271.

[3] G. Russo and P. Smereka, Kinetic theory for bubbly flows I, II.SIAM J. Appl. Math. 56 (1996) 327–371.

[4] V. Teshukhov, G. Russo, A. Chesnokov, Analytical and numeri-cal solutions of the shallow water equations for 2d rotational flows,Math. Models Methods Appl. Sci. 14 (2004) 1451.

[5] S.P. Tsarev, On Poisson brackets and one-dimensional Hamilto-nian systems of hydrodynamic type, Soviet Math. Dokl., 31 (1985)488–491. S.P. Tsarev, The geometry of Hamiltonian systems of hy-drodynamic type. The generalized hodograph method, Math. USSRIzvestiya, 37 No. 2 (1991) 397–419.


Rigorous existence results for the Euler or Navier-Stokesequations from a posteriori analysis of approximate solutions

Livio Pizzocchero

The Cauchy problem is considered for the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus Td, in the frameworkof the Sobolev spaces Hn(Td) (n > d/2 + 1).

In papers [1][2][3] (partly inspired by, or related to [4][5][6]) an ap-proach has been developed to obtain rigorous and fully quantitativeresults on the exact solution u of the Euler or NS Cauchy problem froma posteriori analysis of any approximate solution ua.

This approach produces estimates on the interval of existence [0, T )of the exact solution u, and on the Sobolev distance between the exactand the approximate solution. The latter estimate has the form ||u(t)−ua(t)||n ≤ Rn(t) where Rn(t) is a real, nonnegative function of time t,obtained solving a differential “control inequality”. In particular, theexact solution u of the Cauchy problem is global in time if the controlinequality has a global solution Rn : [0,+∞) → [0,+∞).

In the present communication, the above general setting is exempli-fied working in dimension d = 3 with simple initial data (such as theBehr-Necas-Wu vortex [7]). The approximate solutions in these exam-ples are of the following types:

(i) Galerkin solutions for Euler or NS equations (corresponding tosuitable sets of Fourier modes) [1];

(ii) large order Taylor expansions in time for the Euler equations [8][9];

(iii) large order expansions in the Reynolds number for the NS equa-tions [10][11].

Under specific conditions (on the datum and/or the viscosity), the ap-proach based on (i) or (iii) allows to infer the global existence of theexact solution u for the NS Cauchy problem. In cases (ii)(iii), the con-struction of the approximate solution and its a posteriori analysis is per-formed using tools for automatic symbolic computation, finally yielding


“computer assisted proofs” of existence and regularity for the Euler orNS equations.

References

[1] C. Morosi, L. Pizzocchero, Nonlinear Analysis 75 (2012), 2209-2235.

[2] C. Morosi, L. Pizzocchero, Commun. Pure Appl. Analysis 11(2012), 557-586.

[3] C. Morosi, L. Pizzocchero, Appl. Math. Lett. 26(2), 277-284(2013)

[4] S.I. Chernyshenko, P. Constantin, J.C. Robinson, E.S. Titi, J.Math. Phys. 48 (2007), 065204.

[5] C. Morosi, L. Pizzocchero, Rev. Math. Phys. 20 (2008), 625-706.

[6] C. Morosi, L. Pizzocchero, Nonlinear Analysis 74 (2011), 2398-2414.

[7] E. Behr, J. Necas, H. Wu, ESAIM: M2AN 35 (2001), 229-238.

[8] C. Morosi, M. Pernici, L. Pizzocchero, ESAIM: MathematicalModelling and Numerical Analysis 47(3) (2013), 663-688.

[9] L. Pizzocchero, contributed talk, Hyp2012, Padova.

[10] C. Morosi, L. Pizzocchero, arXiv:1304.2972v1 [math.AP] 10 Apr2013.

[11] C. Morosi, M. Pernici, L. Pizzocchero, in preparation.


Asymptotic preserving automatic domain decompositionforthe Vlasov-Poisson-BGK system with applications to

plasmas

Vittorio Rispoli

Results presented in this contribution are the product of an ongoingwork in collaboration with Giacomo Dimarco (University of Toulouse,France) and Luc Mieussens (University of Bordeaux). Our target is todevelop a general numerical method for the solution of plasma physicsmodels. We aim to design a scheme which is high order accurate and, atthe same time, efficient in terms of computational and memory storagecosts, in order to be used for real life applications. Indeed, both these re-quirements are of paramount importance for realistic simulations whichare, in general, big dimensional problems.

The most widely used models for plasma physics are both fluid sys-tems, such as compressible Euler or Navier-Stokes equations, and kineticequations, such as those of Boltzmann or Vlasov type. Latter descrip-tions are more precise in representing the physics of various phenomenabut unfortunately solving kinetic equations has, in general, prohibitivecost in terms of computational power and memory storage. Typical ex-amples of applications we wish to study include simulations of plasmasproduced around hypersonic bodies, ion wind of corona discharges andmagnetic fusion processes.

Plasma dynamics is characterized by a wide range of spatial and tem-poral scales. Depending on conditions, macroscopic Euler-like systemsor kinetic equations of Boltzmann type are commonly used for describ-ing these problems. The most widely used kinetic model is a systemof Vlasov equations describing the dinamics of the particles constitut-ing the plasma. The system is usually coupled with a Poisson equationfor the electric potential or, more generally, through Maxwell equationsto include also magnetic and electromagnetic effects. When collisionsbetween particles are also taken into account, we refer to it as the colli-sional Vlasov model. In this work we consider this last kind of problems,modeling collisions by a relaxation towards the thermodynamical equi-


librium and we refer to such description as the Vlasov-BGK model. Itis well known that, when sufficiently close to equilibrium, kinetic de-scriptions can be replaced by Euler or Navier-Stokes based models, alsocoupled with Poisson or Maxwell equations. Even if fluid models aresufficiently accurate to describe many observed phenomena, for some ofthem this choice is inadequate and in such cases, it turns out that it isstrictly necessary to consider a kinetic description to correctly representsolutions.

In many situations solving the kinetic equations is required only insmall zones where the departure from thermodynamical equilibrium islarge, like near shock layers or where the electric field is strong, whileinside the rest of the computational domain the microscopic descriptionis unnecessary because a fluid system would also provide sufficientlyaccurate solutions. In this work we precisely address such situation inwhich it is possible to exploit the advantages of Domain Decompositiontechniques, carefully and properly discretizing the derived system. Thegeneral framework for the construction of hybrid kinetic-fluid schemesinvolves three main problems: the first one is how to define propercriteria in order to identify the kinetic and the fluid regions, the secondmain problem is how to mathematically realize the coupling and thethird one is the choice of the appropriate numerical methods for solvingthe macroscopic and the microscopic equations.

A possible strategy for the implementation of domain decomposi-tion methods is based on the introduction of a buffer zone in which thetransition from one model to the other and vice-versa is gradual. There-fore, inside buffer zones both models are solved and the solution of thefull problem is obtained as the combination of the kinetic and of thefluid solutions. The introduction of the intermediate zone makes eachof the models degenerate at the borders and so in this way no interfaceconditions are needed. The decomposition of the domain is dynamicin the sense that it applies everywhere in the domain the suited modelfollowing the time evolution of the system. For this reason we consideran evolution equation also for the coupling function.

The second key point of the proposed method is the choice of thenumerical scheme. Concerning the time dicretization we adopt the so-


called Asymptotic Preserving (AP) and Asymptotically Accurate (AA)schemes for solving the Vlasov-BGK equation. This is very importantsince we are considering at the same time different regimes in differentspatial regions and thus we may have to solve the Vlasov-BGK equa-tion close to the fluid regime, which is usually a real challenge if generalnumerical methods are used. In dense regimes, in fact, the collision rategrows exponentially and the collisional time becomes very small. In suchsituations, standard computational approaches lose their efficiency dueto the necessity of using very small time steps in deterministic schemesor, equivalently, a large number of collisions in probabilistic approaches.On the other hand, the time scale of the evolution is the slower fluiddynamic one, which can be much larger than that determined from thecollisional operator and we would like to relate our time step restrictiononly to the slow scale. To solve this problem we choose to constructour method using AP and AA discretizations for the collisional Vlasovequation. These schemes automatically reduce to a consistent numeri-cal approximations of the relevant macroscopic model when the scalingparameter of the system goes to zero. Thanks to the use of this strategywe solve the kinetic equation, in the regions of the domain where it isneeded, using a discretization which is consistent also with that for themacroscopic model and this is fundamental for the correct coupling ofthe two solvers.

In order to achieve also high computational efficiency we adopt aproper numerical strategy, thanks to which we overcome the strongtime step restriction caused by the stiff character of the collision term.In practice we use an IMplicit-Explicit (IMEX) strategy, computingthe kinetic collisional operator with the implicit solver and all the restwith the explicit one. In this way we are able to greatly reduce theglobal computational cost of the simulations and, thanks to the AP andAA properties, we still achieve high resolution in time and phase spaceindependently on the regime. Moreover, we implement and analyze amemory saving strategy in order to make our scheme really efficient alsoregarding storage issues.

Numerical tests are developed, up to now, in a one space dimensionand three velocity dimensions setting. The first extension we plan to


do is to perform simulations in two and three space dimensions. Theobtained results show good accuracy and high computational efficiencyand are a promising basis for the upcoming work.


A Mixed System Inspired by Biology

Elena Rossi

This talk is centered on a new model consisting of a balance law anda parabolic equation aiming at the description of two competing pop-ulations, described by their densities u and w. While w diffuses, upropagates towards the regions where w is highest. A key parameter isthe horizon at which u feels the presence of w.

The resulting system is proved to be well posed in Rn with respect

to the L1 topology. Various numerical integrations show the varietyof the patterns arising in the solutions to this system: depending onthe choice of the various parameters, apparently periodic fluctuationsor asymptotically stable states arise.

This work is in collaboration with R.M. Colombo.


[SS1] On a quasilinear hyperbolic system relative to anatmospheric model on the transition of water defined on the

whole space.+

[SS2] An initial boundary value problem on a strip for aquasilinear hyperbolic system relative to a model of

H2O-phase transitions in atmosphere.

Steave C. Selvaduray

In [2], a model of the phase transition of H2O in the three states inatmosphere is proposed. The unknow functions in the model are thedensity ρ of dry air, the density π of water vapour, the density σl ofwater in the liquid state, the density σs of water in the solid state, thespeed v and the temperature T of the atmospheric gas; they are linkedtogether through the continuity equations, the equation of momentumof the gas and the equation of energy balance. Furthermore the speed ul

of droplets and the speed us of ice crystalls are given by simple formulasinvolving v. In [2], we have obtained a theorem of existence and unique-

ness of the solution (ρ, π, σl, σs, v, T ) in the class C0(

[0, τ ] ;W 1p (Ω)

)2 ×C0

(

[0, τ ] ;W 1p (Ω× R+)

)2 ×W 2,1p ([0, τ ]× Ω)×W 2,1

q ([0, τ ]× Ω)(3 < q < p < 2q, 4 < p < ∞; τ > 0 sufficiently small), where, apart fromtechnical assumptions about initial and boundary conditions, the equa-tions of the model are defined on bounded open set Ω, the speed v andthe external force are supposed to be tangent to the boundary of Ω;moreover, to avoid difficulties, the density π has been replaced by itslocal average in the equations for σl, σs.

In [1], we have studied the hyperbolic part of the model in [2] on abounded domain Ω, with fixed velocities v, ul, us and temperature T ,supposing that v, ul, us are tangential to the boundary of Ω. Under thatconditions, without introduction of local average of π, we have proved atheorem of existence and uniqueness of the solution (ρ, π, σl, σs) in the

class L∞(

0, τ ;W 1∞ (Ω)

)2×L∞(

0, τ ;W 1∞ (Ω× R+)

)2∩W 1q (0, τ ;L∞ (Ω))

2

×W 1q (0, τ ;L∞ (Ω× R+))

2(q ∈ [1,∞], τ > 0 sufficiently small) with

continuous dependence on the data.


In the manuscript [SS1], we study the Cauchy problem for the hy-perbolic part of the model [2] in the whole space Ω = R

3; this set ofequations is a diagonal quasilinear hyperbolic system containing inte-gral quadratic and Lipschitz continuous terms. The process to prove thewellposedness, in the Lipschitz class, for this system, is an extension ofthe method of characteristics inaugurated by M. Cinquini-Cibrario.

In the manuscript [SS2] we study the hyperbolic part of the modelin [2] on the strip Ω = R

2 × (0, 1) with fixed velocities v, ul, us, andtemperature T . Moreover, we assume v to be tangent to the planesz = 0 and z = 1, whereas ul and us have negative vertical compo-nents; therefore, rain and ice fall from the strip. To the equationswe associate initial conditions for ρ, π and boundary conditions onΓ− = [0 × Ω× R+] ∪

[

[0, t1)× R2 × 1 × R+

]

for σl, σs. Further-more, we ask that

ul, us ∈ L∞ ((0, τ)× Ω× R+)3 ∩ L1

(

0, τ ;W 1∞(Ω× R+)

)3

ulz, usz ∈ L1z

(

0, 1;W 1∞((0, τ)× R

2 × R+))

∇ · ul,∇ · us ∈ L1(

0, τ ;W 1∞(Ω× R+)

)

.

To prove the wellposedness of this problem, we extend the method ofcharacteristics used in [SS1]. Indeed we replace the initial boundary-value problem by its system of characteristics in integral form.

[1] Ascoli D., Selvaduray, S. C.: Wellposedness in the Lipschitz class

for a hyperbolic system arising from a model of the atmosphere in-

cluding water phase transitions. Quaderno Dip. Mat. Univ. Torinon.6/2012. (To appear on NoDEA.)

[2] Selvaduray, S. C., Fujita Yashima H.: Equazioni del moto dell’aria

con la transizione di fase dell’acqua nei tre stati: gassoso, liquido

e solido. Memorie della classe di scienze Fisiche, Matematiche eNaturali, Serie V, Volume 35, pp.37-69, Accademia delle Scienze diTorino, 2011.


Initial-boundary value problems for continuity equationswith low regularity coefficients

Laura V. Spinolo

The analysis of continuity equations with weakly differentiable coeffi-cients has recently received considerable attention. In particular, thefundamental papers by Di Perna and Lions and by Ambrosio establishexistence and uniqueness of solutions of the Cauchy problem for con-tinuity equations with Sobolev and BV (bounded total variation) coef-ficients, respectively. This analysis has important applications to thestudy of several nonlinear partial differential equations, like hyperbolicsystems of conservation laws in several space dimensions.

In this talk I will discuss recent results establishing existence anduniqueness of solutions of initial-boundary value problems for continuityequations with BV (bounded total variation) coefficients. I will alsoexhibit some new counter-examples showing that, as soon as the BVregularity deteriorates at the domain boundary, uniqueness is in generalviolated.

The talk will be based on joint works with G. Crippa and C. Dona-dello.


Time-delayed instabilities in complex Burgers equations

Marta Strani

For Burgers equations with real data and complex forcing terms, N.Lerner, Y. Morimoto and C. J. Xu [1] proved that only analytical datagenerate local C2 solutions. These instabilities are however not ob-served numerically; rather, numerical simulations show an exponentialgrowth only after a delay in time. We argue that numerical diffusionis responsible for this time delay, and we show that the introduction ofa small O(ε) viscous term in the equation can imply uniform boundsin time O(

√ε). In particular, we consider the Cauchy problem for the

viscous complex Burgers equations in the torus with small viscosity oforder O(ε), and we show that initial data u0(x) = a (x/ε) with largefrequencies O(1/ε) generate solutions that are bounded in time O(1),before exhibiting an exponential growth in time.

These results have been obtained in [2] in collaboration with BenjaminTexier.

[1] Lerner N., Morimoto Y., Xu C. J., Instability of the

Cauchy-Kovalevskaya solution for a class of nonlinear systemsAmer.J. Math. 132 (2010), no 1, 99–123

[2] Strani M., Texier B., Time-delayed instabilities in complex Burgersequations preprint.


Contact discontinuities in 2D compressible MHD

Paola Trebeschi

We study the free boundary problem for contact discontinuities for com-pressible magneto-hydrodynamics (MHD). Under a stability conditionwe derive an a priori estimate in the Sobolev space for the linearizedproblem around some non constant basic state. The result obtained isa joint work with Alessandro Morando and Yuri Trakhinin.

Documents

IperMiB2013 - peano.matapp.unimib.itpeano.matapp.unimib.it/ipermib2013/f/BookOfAbstracts.pdfIperMiB2013 15th Italian Meeting ... Dipartimento di Matematica e Applicazioni Università