on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

hyperbolic structure for a simplified model of dynamical perfect plasticity this paper is devoted to confronting two different approaches to the problem of dynamical perfect plasticity interpreting this model as a constrained boundary value friedrichs system enables one to derive admissible hyperbolic boundary conditions using variational methods we show the well posedness of this problem in a suitable weak measure theoretical setting thanks to the property of finite speed propagation we establish a new regularity result for the solution in short time finally we prove that this variational solution is actually a solution of the hyperbolic formulation in a suitable dissipative entropic sense and that a partial converse statement holds under an additional time regularity assumption for the dissipative solutions

a well balanced scheme for the shallow water equations with topography a non negativity preserving and well balanced scheme that exactly preserves all the smooth steady states of the shallow water system including the moving ones is proposed in addition the scheme must deal with vanishing water heights and transitions between wet and dry areas a godunov type method is derived by using a relevant average of the source terms within the scheme in order to enforce the required well balance property a second order well balanced muscl extension is also designed numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states

quasi newton methods for implicit black box fsi coupling abstract in this paper we introduce a new multi vector update quasi newton mvqn method for implicit coupling of partitioned transient fsi solvers the new quasi newton method facilitates the use of black box field solvers and under certain circumstances can be demonstrated to provide newton like convergence behaviour for strongly coupled fsi benchmark problems we demonstrate the superior convergence behaviour and robust nature of the mvqn method compared to other well known quasi newton coupling schemes including the least squares reduced order modelling ibqn ls scheme the classical rank 1 update broyden s method and fixed point iterations with dynamic relaxation the quasi newton methods are analysed on a suite of strongly coupled fsi problems including but not limited to internal incompressible flow through a flexible tube where the solid density is an order of magnitude lower than the fluid density

convergence of relaxation schemes for initial boundary value problems for conservation laws abstract we consider a scalar conservation law with stiff source term in the quarter plan this equation is relaxed in a quasi linear hyperbolic system the relaxed system is approximated using an upwind scheme converging for fixed relaxation time and vanishing discretization mesh further we use chapman enskog expansion to obtain a viscous scheme for the equilibrium law and prove its convergence to the physical solution boundary information is carefully handled in an appropriate inequality linking the entropy numerical flux and the flux function an extension to general conservative schemes is also investigated

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation abstract we consider a kinetic bgk model relaxing to isentropic gas dynamics previously introduced by the authors but with dirichlet boundary condition on the incoming velocities we pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes our method is very general and could be applied to any entropy satisfying bgk model as soon as we have strong compactness of the macroscopic variables inside the domain

rational function certification of multisum integral q identities the method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi integral sums and q integral sums

on the controllability of the non isentropic 1 d euler equation in this paper we examine the question of the boundary controllability of the one dimensional non isentropic euler equation for compressible polytropic gas in the context of weak entropy solutions we consider the system in eulerian coordinates and the one in lagrangian coordinates we obtain for both systems a result of controllability toward constant states with the limitation gamma frac 5 3 on the adiabatic constant for the lagrangian system the solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

well posedness of general boundary value problems for scalar conservation laws in this paper we investigate well posedness for the problem u t div ph u f on 0 t times om om subset r n with initial condition u 0 cdot u 0 on om and with general dissipative boundary conditions varphi u cdot nu in beta t x u on 0 t times ptl om here for a e t x in 0 t times ptl om beta t x cdot is a maximal monotone graph on r this includes as particular cases dirichlet neumann robin obstacle boundary conditions and their piecewise combinations as for the well studied case of the dirichlet condition one has to interprete the it formal boundary condition given by beta by replacing it with the adequate it effective boundary condition such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation we claim that the formal boundary condition given by beta should be interpreted as the effective boundary condition given by another monotone graph tilde beta which is defined from beta by the projection procedure we describe we give several equivalent definitions of entropy solutions associated with tilde beta and thus also with beta for the notion of solution defined in this way we prove existence uniqueness and l 1 contraction monotone and continuous dependence on the graph beta convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results

extending the robustness and efficiency of artificial compressibility for partitioned fluid structure interactions abstract in this paper we introduce the idea of combining artificial compressibility ac with quasi newton qn methods to solve strongly coupled fully quasi enclosed fluid structure interaction fsi problems partitioned incompressible fsi based on dirichlet neumann domain decomposition solution schemes cannot be applied to problems where the fluid domain is fully enclosed a simple example often provided in literature is that of a balloon with a prescribed inflow velocity in this context artificial compressibility ac is a useful method by which the incompressibility constraint can be relaxed by including a source term within the fluid continuity equation the attractiveness of ac stems from the fact that this source term can readily be added to almost any fluid field solver including most commercial solvers once included both the modified fluid solver and structural solver can be treated as black box field operators ac is however limited to the class of problems it can effectively be applied to for example ac is an efficient solution strategy for the simulation of blood flow through arteries but performs poorly when applied to the simulation of blood flow through an opening heart valve the focus of this paper is thus to extend the application of ac by including an additional newton system accounting for the missing interface sensitivities we do so through the use of a multi vector update quasi newton mvqn method where the required system jacobians are approximated rather than explicitly computed in so doing we continue to facilitate the notion that the ac modified fluid field solver and solid field solver can be treated as black box solvers we aim to demonstrate the improved performance of the combination of ac qn when compared to ac applied in isolation

ecoulements de fluides parfaits en deux variables independantes de type espace reflexion d un choc plan par un diedre compressif we consider the euler equations of a perfect fluid having only two independent space like variables which account for the stationary 2 dimensional or axisymmetrical 3 dimensional cases as well as the 2 dimensional riemann problem we show that the pressure and the angle between an axis and the velocity field satisfy a first order system which turns out to be elliptic in the subsonic zone in particular the pressure satisfies a maximum principle which has not been stated before to the best of my knowledge using this and the bernouilli law we give various a priori estimates of the pressure the density the enthalpy and the velocity in the problem of the reflection of a shock wave by a wedge we also bound the size of the subsonic region and the force that the fluid applies to the boundary

on the initial boundary value problem for temple systems this paper is devoted to the proof of the well posedness of the initial boundary value problem for a system of conservation laws of temple type no assumption on the monotonicity of the eigenvalues along the lax curves is assumed and the total variation of the initial data is not required to be small the solution is shown to be a lipschitz function of the initial and boundary data of time and of the boundary profile as well

a q analogue of the j 2 supercongruence of van hamme abstract the l 2 supercongruence of van hamme was proved by swisher recently in this paper we provide a conjectural q analogue of the l 2 supercongruence of van hamme and prove a weaker form of it by using the q wz method in the same way we prove a complete q analogue of the following congruence k 0 n 6 k 1 2 k k 3 512 n k 0 mod 4 2 n 1 2 n n which was conjectured by z w sun and confirmed by b he we also provide a conjectural q analogue of another congruence proved by swisher

the boundary riemann solver coming from the real vanishing viscosity approximation we study the limit of the hyperbolic parabolic approximation left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr tilde rm ss v varepsilon t 0 equiv bar g cr v varepsilon 0 x equiv bar v 0 end array right the function tilde ss is defined in such a way as to guarantee that the initial boundary value problem is well posed even if tilde b is not invertible the data bar g and bar v 0 are constant when tilde b is invertible the previous problem takes the simpler form left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr v varepsilon t 0 equiv bar v b cr v varepsilon 0 x equiv bar v 0 end array right again the data bar v b and bar v 0 are constant the conservative case is included in the previous formulations convergence of the v varepsilon smallness of the total variation and other technical hypotheses are assumed and a complete characterization of the limit is provided the most interesting points are the following first the boundary characteristic case is considered that is one eigenvalue of tilde a can be 0 second as pointed out before we take into account the possibility that tilde b is not invertible to deal with this case we take as hypotheses conditions that were introduced by kawashima and shizuta relying on physically meaningful examples we also introduce a new condition of block linear degeneracy we prove that if this condition is not satisfied then pathological behaviors may occur

continuous dependence for 2 2 conservation laws with boundary on the domain 0 t x r t 0 and x 9 t for some boundary profile 9 r r as usual 1 1 is assumed to be strictly hyperbolic and with each characteristic field either linearly degenerate or genuinely non linear an initial data u 0 x u x having sufficiently small total variation is given we consider two different kinds of boundary conditions along x 9 t and in both cases we construct a lipschitzean flow whose trajectories are solutions of an initial boundary value problem for 1 1 thus we prove the continuous dependence of the solution upon the initial data upon the boundary condition and upon the boundary profile the existence theory for global bv solutions to the cauchy problem for 1 1 goes back to the fundamental paper 15 by glimm more recently in 6 a new approach has been introduced it relies on the construction of a lipschitzean semigroup the standard riemann semigroup srs whose trajectories extend the local standard lax 17 solutions of riemann problems at present such a srs has been constructed in the 2 2 case in 8 9 while for the general n n case see 7 10 and 3 in article no de973274

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

entropy satisfying relaxation method with large time steps for euler ibvps this paper could have been given the title how to positively and implicitly solve euler equations using only linear scalar advections the new relaxation method we propose is able to solve euler like systems as well as initial and boundary value problems with real state laws at very low cost using a hybrid explicit implicit time integration associated with the arbitrary lagrangian eulerian formalism furthermore it possesses many attractive properties such as i the preservation of positivity for densities ii the guarantee of min max principle for mass fractions iii the satisfaction of entropy inequality under an expressible bound on the cfl ratio the main feature that will be emphasized is the design of this optimal time step which takes into account data not only from the inner domain but also from the boundary conditions

etude variationnelle du probleme de riemann abstract we study the riemann problem for a system of conservation laws with a strongly convex entropy in arbitrary dimensions we show that the resolution of this problem can be reduced to solving an infinite number of entropy inequalities after this transformation we consider various applications maximum principles uniqueness of the solution of the trivial riemann problem etc these inequalities are also used for a study of the wave curves in connection with the ibvp as considered in a benabdallah et d serre problemes aux limites pour des systemes hyperboliques non lineaires de deux equations a une dimension d espace c r acad sci paris ser i math 305 1987 677 680 f dubois and p le floch boundary conditions for nonlinear hyperbolic system of conservation laws j differential equations71 1988 93 122 at last we propose a numerical scheme based on the discretization of the entropy inequalities for the solution of the monodimensional riemann problem wave curves are not involved thus multi dimensional extensions can be expected

on general balance laws with boundary this paper is devoted to general balance laws with a possibly non local source term with a non characteristic boundary basic well posedness results are obtained trying to provide sharp estimates in particular bounds tend to blow up as the boundary tends to be characteristic new uniqueness results for the solutions to conservation and or balance laws with boundary are also provided

blow up of solutions to the initial boundary value problem for quasilinear hyperbolic systems of conservation laws abstract this work is a continuation of our previous work z q shao d x kong y c li shock reflection for general quasilinear hyperbolic systems of conservation laws nonlinear anal 66 1 2007 93 124 in the present paper we study the global structure instability of the self similar solution u u x t containing shocks contact discontinuities and at least one rarefaction wave of the initial boundary riemann problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane t x t 0 x 0 we prove the nonexistence of global piecewise c 1 solution to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane t x t 0 x 0 our result indicates that the kind of riemann solution u u x t mentioned above is globally structurally unstable as an application of our result we consider the model proposed by aw and rascle a aw m rascle resurrection of second order models of traffic flow siam j appl math 60 2000 916 938 describing traffic flow on a road network following the work of garavello and piccoli m garavello b piccoli traffic flow on a road network using the aw rascle model comm partial differential equations 31 2 2006 243 275 we prove the nonexistence of global piecewise c 1 solution containing one rarefaction wave of the initial boundary value problem for this model

stability of the 1d ibvp for a non autonomous scalar conservation law we prove the stability with respect to the flux of solutions to initial boundary value problems for scalar non autonomous conservation laws in one space dimension key estimates are obtained through a careful construction of the solutions

symmetrization of dissipative dispersive traveling waves for systems of conservation laws abstract this work addresses existence of traveling waves for systems of conservation laws through an hamiltonian formalism a related hamilton jacobi equation is given and studied our proofs rely on the construction of a cost functional and require a smallness condition for the dispersion with respect to diffusion some numerical applications are also presented concerning a model phase transition problem

the numerical interface coupling of nonlinear hyperbolic systems of conservation laws i the scalar case we study the theoretical and numerical coupling of two general hyperbolic conservation laws the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model in order to analyze the convergence of the coupled numerical scheme we first revisit the approximation of the boundary value problems we then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations the general coupling problem is analyzed for riemann initial data and illustrated by numerical simulations resume nous nous interessons a une nouvelle forme de couplage de deux systemes hyperboliques de lois de conservation ce couplage assure de facon faible la continuite de la solution a l interface sans imposer la conservativite du modele couple pour etudier la convergence du schema d approximation numerique nous commencons par reprendre les resultats concernant l approximation du probleme aux limites nous demontrons ensuite la convergence du schema couple dans un certain nombre de cas interessants le cas general du couplage est etudie et illustre numeriquement pour une donnee initiale de riemann

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data we consider the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic conservation laws in particular we consider initial boundary value problems where the initial and boundary data switch between different c 1 functions at certain switching points and both the functions and the switching points are controlled we show that the control to state mapping is differentiable in a certain generalized sense which implies frechet differentiability with respect to the control functions and the switching points for the composition with a tracking type functional even in the presence of shocks we also present an adjoint based formula for the gradient of the reduced objective functional

traffic flow modelling with junctions abstract motivated by the modelling of a roundabout we are led to study the traffic on a road with points of entry and exit in this note we would like to describe the modellisation of a junction and solve the riemann problem for such a model more precisely between each point of discontinuity we use a multi class extension of the lwr model to describe the evolution of the density of the vehicles the multi class approach being used in order to distinguish the vehicles after their origin and destination then we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles in the case of the one t road we obtain a result of existence and uniqueness this first step allows us to obtain a similar result for the n t road we describe these results and also some properties of the obtained solutions in order to see how long this model is valid

on the attainable set for temple class systems with boundary controls consider the initial boundary value problem for a strictly hyperbolic genuinely nonlinear temple class system of conservation laws begin equation label eq 1 u t f u x 0 qquad u 0 x ov u x qquad left begin array ll u t a widetilde u a t 2pt u t b widetilde u b t end array right end equation on the domain omega t x in r 2 t geq 0 a le x leq b we study the mixed problem ref eq 1 from the point of view of control theory taking the initial data overline u fixed and regarding the boundary data widetilde u a widetilde u b as control functions that vary in prescribed sets u a u b of li boundary controls in particular we consider the family of configurations a t doteq u t cdot u mbox is a sol to ref eq 1 widetilde u a in u a widetilde u b in u b that can be attained by the system at a given time t 0 and we give a description of the attainable set a t in terms of suitable oleinik type conditions we also establish closure and compactness of the set a t in the lu topology

well posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition in this paper we give a simple proof of well posedness of multidimensional scalar conservations laws with a strong boundary condition the proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation

hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries we consider attractive particle systems in mathbb z d with product invariant measures we prove that when particles are restricted to a subset of mathbb z d with birth and death dynamics at the boundaries the hydrodynamic limit is given by the unique entropy solution of a conservation law with boundary conditions in the sense of bardos et al comm part diff equ 4 1017 1034 1979 for the hydrostatic limit between parallel hyperplanes we prove a multidimensional version of the phase diagram conjectured in popkov and schutz europhys lett 48 257 263 1999 and show that it is robust with respect to perturbations of the boundaries

scalar multidimensional conservation laws ibvp in noncylindrical lipschitz domains we study the initial boundary value problems for multidimensional scalar conservation laws in noncylindrical domains with lipschitz boundary we show the existenceuniqueness of this problem for initial boundary data in l 1 and the flux function in the class c 1 in fact first considering smooth boundary we obtain the l 1 contraction property discuss the existence problem and prove it by the young measures theory in the end we show how to pass the existence uniqueness results on to some domains with lipschitz boundary

fluid structure partitioned procedures based on robin transmission conditions in this article we design new partitioned procedures for fluid structure interaction problems based on robin type transmission conditions the choice of the coefficient in the robin conditions is justified via simplified models the strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane as in hemodynamics applications we analyze theoretically the new iterative procedures on a model problem which represents a simplified blood vessel system in particular the robin neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures the theoretical results are checked using numerical experimentation

the initial boundary value problem for general non local scalar conservation laws in one space dimension we prove global well posedness results for weak entropy solutions of bounded variation bv of scalar conservation laws with non local flux on bounded domains under suitable regularity assumptions on the flux function in particular existence is obtained by proving the convergence of an adapted lax friedrichs algorithm lipschitz continuos dependence from initial and boundary data is derived applying kru hkov s doubling of variable technique

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

inverse problem for non viscous mean field control example from traffic this paper presents an inverse problem for mean field games where we find the mean field problem statement for which the given dynamics is the solution we use distributed traffic as an example and derive the classic lighthill whitham richards lwr model as a solution of the non viscous mean field game we also derive the same model by choosing a different problem where we use travel time which is a distributed parameter as the cost for the optimal control we then study the stationary versions of these two problems and provide numerical solutions for the same

initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws we consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case the entropy solutions we address assume their initial data only in the sense of weak star in l as t 0 and satisfy the entropy inequality in the sense of distributions for t 0 we prove that if the flux function has weakly genuine nonlinearity then the entropy solutions are always unique and the initial layers do not appear we also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation

coupling techniques for nonlinear hyperbolic equations iv well balanced schemes for scalar multi dimensional and multi component laws this series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations the coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations in an earlier work this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations in the present paper we significantly extend this framework and in addition encompass equations in several space variables this new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space our main contributions are on one hand the design and analysis of a well balanced finite volume method on general triangulations and on the other hand a proof of convergence of this method toward entropy solutions extending coquel cockburn and lefloch s theory restricted to a single conservation law without coupling the core of our analysis is first the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and second a proof of convergence toward the entropy solution of the coupling problem

macdonald s constant term conjectures for exceptional root systems we announce proofs of macdonald s constant term conjectures for the affine root systems s f 4 and s f 4 v we also give an algorithm for deciding the conjectures for the remaining root systems s e 6 s e 7 and s e 8 and prove that the constant term in question can indeed be expressed in closed form combined with previous work of zeilberger bressoud kadell and gustafson our results imply that macdonald s conjectures are true in form for any root system

global structure stability of riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary j math anal appl 330 1 2007 511 540 in the present paper we study the global structure stability of the riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the existence and uniqueness of global piecewise c 1 solution containing only shocks and contact discontinuities to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws in the half space t x t 0 x 0 our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in a half space possesses a global structural stability some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines are also given

initial boundary value problems for conservation laws with source terms and the degasperis procesi equation we consider conservation laws with source terms in a bounded domain with dirichlet boundary conditions we first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition equipped with this formulation we go on to establish the well posedness of entropy solutions to the initial boundary value problem the proof utilizes the kinetic formulation and the averaging lemma finally we make use of these results to demonstrate the well posedness in a class of discontinuous solutions to the initial boundary value problem for the degasperis procesi shallow water equation which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term

existence in the large for riemann problems for systems of conservation laws on obtient un theoreme d existence pour le probleme de riemann pour des systemes non lineaires de lois de conservation on utilise l inegalite d entropie pour obtenir une estimation a priori des forces des chocs et des ondes de refraction formant une solution il s ensuit l existence d une telle solution par une application de la theorie du degre de dimension finie

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

multi dimensional initial boundary value problems with strong nonlinearities initial boundary value problems for the general scalar singularly perturbed parabolic equation in a cylindrical domain are studied the one dimensional reaction diffusion equation is considered in the absence of convection then the general one dimensional and n dimensional reaction convection diffusion equations are analyzed

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

a well posed conservation law with a variable unilateral constraint this paper considers the cauchy problem for a conservation law with a variable unilateral constraint its motivation being for instance the modeling of a toll gate along a highway this problem is solved by means of nonclassical shocks and its well posedness is proved then the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem

explicit formula for the limit of a difference approximation we consider the conservation law u t log ae u be u x 0 where a 0 b 0 a b 1 with initial condition u x 0 u o x in the quarter plane x 0 t 0 we are interested in studying the limit of the approximate solution defined by lax s difference scheme and we discuss the problem of percribing u 0 t t arbitrarily

kinetic schemes on staggered grids for barotropic euler models entropy stability analysis we introduce in the one dimensional framework a new scheme of finite volume type for barotropic euler equations the numerical unknowns namely densities and velocities are defined on staggered grids the numerical fluxes are defined by using the framework of kinetic schemes we can consider general convex pressure laws we justify that the density remains non negative and the total physical entropy does not increase under suitable stability conditions performances of the scheme are illustrated through a set of numerical experiments

boundary conditions for scalar conservation laws from a kinetic point of view boundary conditions for multidimensional scalar conservation laws are obtained in the context of hydrodynamic limits from a kinetic point of view the initial boundary value kinetic problem is well posed since inward and outward characteristics of the domain can be distinguished the convergence of the first momentum of the distribution function to an entropy solution of the conservation law is established boundary conditions are obtained the equivalence with the bardos leroux and nedelec conditions is studied

formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary we study nonlinear hyperbolic conservation laws posed on a differential n 1 manifold with boundary referred to as a spacetime and defined from a prescribed flux field of n forms depending on a parameter the unknown variable a class of equations proposed by lefloch and okutmustur far east j math sci 31 49 83 2008 our main result is a proof of the convergence of the finite volume method for weak solutions satisfying suitable entropy inequalities a main difference with previous work is that we allow for slices with a boundary and in addition introduce a new formulation of the finite volume method involving the notion of total flux functions under a natural global hyperbolicity condition on the flux field and the spacetime and by assuming that the spacetime admits a foliation by compact slices with boundary we establish an existence and uniqueness theory for the initial and boundary value problem and we prove a contraction property in a geometrically natural l 1 type distance

global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao d x kong y c li shock reflection for general quasilinear hyperbolic systems of conservation laws nonlinear anal tma 66 1 2007 93 124 in this paper we study the global structure instability of the riemann solution u u x t containing shocks at least one rarefaction wave for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the nonexistence of global piecewise c 1 solution to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given

well posedness of balance laws with boundary consider the initial boundary value problem for a temple system of balance laws aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small

on the attainable set for scalar nonlinear conservation laws with boundary control we consider the initial value problem with boundary control for a scalar nonlinear conservation law begin equation u t f u x 0 qquad qquad u 0 x 0 qquad u cdot 0 tilde u in cal u tag ast end equation on the domain omega t x in real 2 t geq 0 x geq 0 here u u t x is the state variable cal u is a set of bounded boundary data regarded as controls and f is assumed to be strictly convex we give a characterization of the set of attainable profiles at a fixed time t 0 and at a fixed point bar x 0 begin equation begin aligned rag u t cdot u hbox is a solution of ast ragx u cdot bar x u hbox is a solution of ast end aligned qquad quad cal u ellein bbb r end equation moreover we prove that rag and ragx are compact subsets of elleuno and elleuno loc respectively whenever cal u is a set of controls which pointwise satisfy closed convex constraints together with some additional integral inequalities

kinetic approximation of a boundary value problem for conservation laws we design numerical schemes for systems of conservation laws with boundary conditions these schemes are based on relaxation approximations taking the form of discrete bgk models with kinetic boundary conditions the resulting schemes are riemann solver free and easily extendable to higher order in time or in space for scalar equations convergence is proved we show numerical examples including solutions of euler equations

a conservation law with point source and discontinuous flux function modelling continuous sedimentation continuous sedimentation of solid particles in a liquid takes place in a clarifier thickener unit which has one feed inlet and two outlets the aim of the paper is to formulate and analyse a dynamic model for this process under idealized physical conditions the conservation of mass yields the scalar conservation law frac partial u x t partial t frac partial partial x f u x t x s t delta x where delta is the dirac measure s is a source and f is a flux function which is discontinuous at three points in the one dimensional space coordinate x under certain regularity assumptions a procedure for constructing a solution locally in time is presented the nonlinear phenomena are complicated and so is the general construction of a solution the problem of nonuniqueness due to the discontinuities of f u cdot is handled by a generalized entropy condition an advantage of this approach is that the a priori boundary conditions at the discontinuities

data assimilation and inverse problem for fluid traffic flow models and algorithms this article deals with traffic data assimilation and algorithms that are able to predict the traffic flow on a road section the traffic flow is modellized by the aw rascle hyperbolic system we have to minimize a functional whose optimization variables are initial condition we use the roe method to compute the solution to the traffic flow modelling system then we compute the gradient of the functional by an adjoint method this gradient will be used to optimize the functional

coupling techniques for nonlinear hyperbolic equations i self similar diffusion for thin interfaces we investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and in particular we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows the main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost as a consequence non uniqueness of weak solutions is observed for the initial value problem and these solutions need to be supplemented with further admissibility conditions this paper is devoted to investigating these issues in the setting of self similar vanishing viscosity approximations to the riemann problem for general hyperbolic systems following earlier works by joseph lefloch and tzavaras we establish an existence theorem for the riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

short note on the similarity between dirichlet neumann with interface artificial compressibility and robin neumann schemes for the solution of fluid structure interaction problems in this note the similarity between two implicit partitioned solution techniques for fluidstructure interaction fsi problems is analyzed using finite volume discretization of the flow equations fluid structure interaction refers to the mutual interaction between a fluid flow and a flexible structure partitioned solution techniques solve the flow equations and the structural equations separately these techniques are classified as implicit or strongly coupled if they satisfy the interaction conditions on the fluid structure interface in each time step and as explicit or loosely coupled if they do not both techniques analyzed in this note use block gauss seidel gs iterations meaning that the flow equations and the structural equations are solved consecutively within a time step until some convergence tolerance is reached as the flow and structural equations are solved separately the interaction conditions on the fluid structure interface have to be converted into boundary conditions on the common boundary of the fluid and structure subdomains several types of boundary conditions can be applied resulting in different decompositions in the case of dirichlet neumann dn decomposition the flow equations are solved with a dirichlet boundary condition given velocity on the fluid structure interface while the structural equations are solved with a neumann boundary condition given stress on the interface conversely robinneumann rn decomposition introduced in 1 denotes a robin boundary condition on the fluid side of the interface and a neumann boundary condition on the structure side the first technique in this comparison is block gauss seidel iterations applied to the monolithic system previously multiplied by a suitable permutation matrix leading to a robin neumann decomposition gs rn this first technique includes a simplified version of the structural model in the flow equations by means of a robin boundary condition to accelerate the convergence of the gs iterations 1 2 the second technique is block gauss seidel iterations with dirichletneumann decomposition and interface artificial compressibility gs dn iac this second