handling solid fluid interfaces for viscous flows explicit jump approximation vs ghost cell approaches the ghost cell approaches gca for handling stationary solid boundaries regular or irregular are first investigated theoretically and numerically for the diffusion equation with dirichlet boundary conditions the main conclusion of this part of investigation is that the approximation for the diffusion term has to be second order accurate everywhere in order for the numerical solution to be rigorously second order accurate violating this principle the linear and quadratic gcas have the following shortcomings 1 restrictive constraints on grid size when the viscosity is small 2 susceptibleness to instability of a time explicit formulation for strongly transient flows 3 convergence deterioration to zeroth or first order for solutions with high frequency modes therefore the widely used linear extrapolation for enforcing no slip boundary conditions should be avoided even for regular solid boundaries as a remedy a simple method based on explicit jump approximation eja is proposed eja hinges on the idea that a velocity derivative jump at the boundary reduces to the value of the velocity derivative at the fluid side because the velocity of the stationary boundary is zero although the time marching linear system of eja is not symmetric it is strictly diagonal dominant with positive diagonal entries numerical results show that over a large range of viscosity and grid sizes eja performs much better than gcas in terms of stability and accuracy furthermore the second order convergence of eja does not depend on viscosity and the spectrum of the solution as those of gcas do this paper is written with enough details so that one can reproduce the numerical results

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a new interface tracking method the polygonal area mapping method we present a new method the polygonal area mapping pam method for tracking a non diffusive immiscible material interface between two materials in two dimensional incompressible flows this method represents material areas explicitly as piecewise polygons traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon clippings in a discrete manner the new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately in addition to high accuracy the pam method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications the mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons the results from a set of widely used benchmark tests show that the pam method is superior to existing volume of fluid vof methods

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

the size of random fragmentation trees we consider the random fragmentation process introduced by kolmogorov where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution the dislocation law this is repeated recursively for all pieces more precisely we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold and we study the associated random tree dean and majumdar found a phase transition for this process the number of fragmentations is asymptotically normal for some dislocation laws but not for others depending on the position of roots of a certain characteristic equation this parallels the behavior of discrete analogues with various random trees that have been studied in computer science we give rigorous proofs of this phase transition and add further details the proof uses the contraction method we extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter we believe that this extension has independent interest

systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation in this paper we systematically derive jump conditions for the immersed interface method siam j numer anal 31 1994 pp 1019 1044 siam j sci comput 18 1997 pp 709 735 to simulate three dimensional incompressible viscous flows subject to moving surfaces the surfaces are represented as singular forces in the navier stokes equations which give rise to discontinuities of flow quantities the principal jump conditions across a closed surface of the velocity the pressure and their normal derivatives have been derived by lai and li appl math lett 14 2001 pp 149 154 in this paper we first extend their derivation to generalized surface parametrization starting from the principal jump conditions we then derive the jump conditions of all first second and third order spatial derivatives of the velocity and the pressure we also derive the jump conditions of first and second order temporal derivatives of the velocity using these jump conditions the immersed interface method is applicable to the simulation of three dimensional incompressible viscous flows subject to moving surfaces where near the surfaces the first and second order spatial derivatives of the velocity and the pressure can be discretized with respectively third and second order accuracy and the first order temporal derivatives of the velocity can be discretized with second order accuracy

fixed points of the smoothing transform two sided solutions given a sequence c t c t1 t2 of real valued random variables with tj 0 for all j 1 and almost surely finite n sup j 1 tj g 0 the smoothing transform associated wi

uniform convergence of martingales in the branching random walk in a discrete time supercritical branching random walk let z n be the point process formed by the n th generation let m lambda be the laplace transform of the intensity measure of z 1 then w n lambda int e lambda x z n dx m lambda n which is the laplace transform of z n normalized by its expected value forms a martingale for any lambda with m lambda finite but nonzero the convergence of these martingales uniformly in lambda for lambda lying in a suitable set is the first main result of this paper this will imply that on that set the martingale limit w lambda is actually an analytic function of lambda the uniform convergence results are used to obtain extensions of known results on the growth of z n nc d with n for bounded intervals d and fixed c this forms the second part of the paper where local large deviation results for z n which are uniform in c are considered finally similar results both on martingale convergence and uniform local large deviations are also obtained for continuous time models including branching brownian motion

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

solutions to complex smoothing equations we consider smoothing equations of the form begin aligned x mathop limits text law sum j ge 1 t j x j c end aligned where c t 1 t 2 ldots is a given sequence of random variables and x 1 x 2 ldots are independent copies of x and independent of the sequence c t 1 t 2 ldots the focus is on complex smoothing equations i e the case where the random variables x c t 1 t 2 ldots are complex valued but also more general multivariate smoothing equations are considered in which the t j are similarity matrices under mild assumptions on c t 1 t 2 ldots we describe the laws of all random variables x solving the above smoothing equation these are the distributions of randomly shifted and stopped levy processes satisfying a certain invariance property called u alpha stability which is related to operator semi stability the results are applied to various examples from applied probability and statistical physics

high order multidimensional and conservative coarse fine interpolation for adaptive mesh refinement abstract the author presents a polynomial based algorithm for high order multidimensional interpolation at the coarse fine interface in the context of adaptive mesh refinement on structured cartesian grids the proposed algorithm reduces coarse fine interpolation to matrix vector products by exploiting the static mesh geometry and a family of nonsingularity preserving stencil transformations as such no linear system is solved at the runtime and the ill conditioning of vandermonde matrix is avoided the algorithm is also generic in that d the dimensionality of the computational domain and p the degree of the interpolating polynomial are both arbitrary positive integers stability and accuracy are verified by interpolating simple functions and by applying the proposed method to adaptively solving poisson s equation and the convection diffusion equation the companion matlab package amrcfi is also freely available for convenience and more implementation details

adaptive tetrahedral meshing in free surface flow an unstructured three dimensional moving mesh algorithm is developed for free surface flows with the capability of simulating large deformation a robust edge swapping algorithm and an optimization based mesh smoothing algorithm are used to improve mesh quality as the interface deforms the edge swapping algorithm uses an optimal swap sequence to transform configurations with several tetrahedra sharing a common edge the optimization based mesh smoothing relocates vertex position so that the minimum quality of the incident cells is maximized also edge contraction and bisection are used to adjust the mesh resolution according to surface curvature these local refinements effectively maintain good mesh quality and appropriate mesh size while avoiding global re meshing this scheme s robustness and accuracy are demonstrated with deforming liquid ligaments periodic jets and colliding droplets

statistical model with a standard γ distribution we study a statistical model consisting of n basic units which interact with each other by exchanging a physical entity according to a given microscopic random law depending on a parameter ensuremath lambda we focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard gamma distribution the model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity ensuremath lambda alternatively from the nature of the equilibrium distribution we show that the model can also be interpreted as a perfect gas at an effective temperature t ensuremath lambda where particles exchange energy in a space with an effective dimension d ensuremath lambda

on donating regions lagrangian flux through a fixed curve lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region a compact point set in which each particle will pass through the curve and contribute to the flux a precise geometric determination of the donating region has not been available until now the author proposes an explicit constructive and analytical definition of the donating region based on two characteristic curves of the flow field viz streaklines and timelines it is also shown that within a time interval of finite size each particle in the proposed donating region has a net effect of going across the fixed curve once the presented analysis might be potentially useful for flow topology studies lagrangian flux calculation and explicit interface tracking

an isochoric domain deformation method for computing steady free surface flows with conserved volumes abstract the domain deformation method has been applied successfully to steady state free surface flows where the volume of the flow domain is unknown v f de almeida gas liquid counterflow through constricted passages ph d thesis university of minnesota minneapolis mn 1995 p a sackinger p r schunk r r rao a newton raphson pseudo solid domain mapping technique for free and moving boundary problems a finite element implementation j comput phys 125 1996 83 103 l c musson two layer slot coating ph d thesis university of minnesota minneapolis mn 2001 however this method does not handle effectively problems where the volume of the flow domain is known a priori this work extends the original domain deformation method to a new isochoric domain deformation method to account for the volume conservation like in the original domain deformation method the unknown shape of the flow domain is mapped onto a reference domain by using the equations of an elastic pseudo solid the difference with the original method is that this pseudo solid is considered incompressible because of the incompressibility the pseudo pressure of the mapping appears as a lagrange multiplier in the equations and it is determined only up to an arbitrary uniform datum by analyzing the coupled fluid flow mapping problem we show that in the finite element setting such pressure datum can be specified by replacing one continuity equation in the fluid domain

a semicoarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients the purpose of this paper is to present a semicoarsening multigrid algorithm for solving the finite difference discretization of symmetric and nonsymmetric two and three dimensional elliptic partial differential equations with highly discontinuous and anisotropic coefficients the discrete equations are assumed to be defined on a logically rectangular grid obtained possibly through grid generation for a problem defined on an irregular domain the basic algorithm is described along with some modifications which are designed to improve its efficiency and robustness for certain types of problem cases fortran codes that implement the two and three dimensional semicoarsening multigrid algorithms are described briefly and numerical results are presented

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

theoretical analysis and simulations of the generalized lotka volterra model the dynamics of generalized lotka volterra systems is studied by theoretical techniques and computer simulations these systems describe the time evolution of the wealth distribution of individuals in a society as well as of the market values of firms in the stock market the individual wealths or market values are given by a set of time dependent variables w i i 1 dots n the equations include a stochastic autocatalytic term representing investments a drift term representing social security payments and a time dependent saturation term due to the finite size of the economy the w i mathrm mathrm s turn out to exhibit a power law distribution of the form p w ensuremath sim w ensuremath 1 ensuremath ensuremath alpha it is shown analytically that the exponent ensuremath alpha can be expressed as a function of one parameter which is the ratio between the constant drift component social security and the fluctuating component investments this result provides a link between the lower and upper cutoffs of this distribution namely between the resources available to the poorest and those available to the richest in a given society the value of ensuremath alpha is found to be insensitive to variations in the saturation term which represent the expansion or contraction of the economy the results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power law distribution with 1l ensuremath alpha l2

hydrodynamics from kinetic models of conservative economies in this paper we introduce and discuss the passage to hydrodynamic equations for kinetic models of conservative economies in which the density of wealth depends on additional parameters like the propensity to invest as in kinetic theory of rarefied gases the closure depends on the knowledge of the homogeneous steady wealth distribution the maxwellian of the underlying kinetic model the collision operator used here is the fokker planck operator introduced by j p bouchaud and m mezard wealth condensation in a simple model of economy physica a 282 2000 536 545 which has been recently obtained in a suitable asymptotic of a boltzmann like model involving both exchanges between agents and speculative trading by s cordier l pareschi and one of the authors s cordier l pareschi g toscani on a kinetic model for a simple market economy j stat phys 120 2005 253 277 numerical simulations on the fluid equations are then proposed and analyzed for various laws of variation of the propensity

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

an adaptive mesh algorithm for evolving surfaces simulation of drop breakup and coalescence abstract an algorithm is presented for the adaptive restructuring of meshes on evolving surfaces the resolution of the relevant local length scale is maintained everywhere with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations the resulting discretization depends on the instantaneous configuration of the surface but is insensitive to the deformation history application of the adaptive discretization algorithm is illustrated with three dimensional boundary integral simulations of deformable drops in stokes flow the results show that the algorithm can accurately resolve detailed features of deformed fluid interfaces including slender filaments associated with drop breakup and dimpled regions associated with drop coalescence our algorithm should be useful in a variety of fields including computational fluid dynamics image processing geographical information systems and biomedical engineering problems

dynamics of money and income distributions we formulate a model of personal income distributions within society our aim is to understand the origin of the natural law first proposed by pareto who suggested that the high end part of the income distribution follows a power law where f 1 w w 1 we generalise a model proposed the so called cc ccm model introducing the possibility of exchanges of money at random times unlike the cc ccm model this generalisation predicts tail exponents greater than unity in line with pareto s prediction and empirical observation

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

identification of asymptotic decay to self similarity for one dimensional filtration equations the objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one dimensional filtration equations it is based on an alternative formulation of the problem using the pseudoinverse of the density s repartition function in particular the numerical approximations can be proven to satisfy a contraction property for a wasserstein metric various numerical results illustrate the ability of this numerical process to capture the time asymptotic decay towards self similar solutions even for fast diffusion equations

seneta heyde norming in the branching random walk malization of the general crump mode jagers branching process is obtained in this case the convergence holds almost surely the results rely heavily on a detailed study of the functional equation that the laplace transform of the limit must satisfy 1 introduction this paper considers the usual supercritical branching random walk thus ignoring the spatial element the population grows like a supercritical galton watson process the initial ancestor is at the origin of the real line and the positions of her children are given by a point process z each of these children has children in the same way in that the positions of each family relative to the parent are given by an independent copy of z and so on individuals are labelled by their line of descent so if u i i 1 n then u is the i th child of the i th child of the i th child of the initial nn 11 ancestor now let u be the generation in which u is born and write v u if v is a strict ancestor of u let t be the set of all people ever born which can be viewed as a tree with the population members as the nodes sums products sets and so on defined with an index ranging over individual s labels will be restricted to those actually born without this being made 4

self similarity and power like tails in nonconservative kinetic models in this paper we discuss the large time behavior of solution of a simple kinetic model of boltzmann maxwell type such that the temperature is time decreasing and or time increasing we show that under the combined effects of the nonlinearity and of the time monotonicity of the temperature the kinetic model has non trivial quasi stationary states with power law tails in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution the same idea is applied to investigate the large time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and or decreasing of the mean wealth in this last case the large time behavior of the solution shows a pareto power law tail numerical results confirm the previous analysis

an improved sph method towards higher order convergence this paper evaluates various formulations of the sph method for solving the euler equations convergence and stability aspects are discussed and tested taking into account subtleties induced by the presence of a free surface the coherence between continuity and momentum equations is considered using a variational study the use of renormalization to improve the accuracy of the simulations is investigated and discussed a new renormalization based formulation involving wide accuracy improvements of the scheme is introduced the classical sph and renormalized approaches are compared to the new method using simple test cases thus outlining the efficiency of this new improved sph method finally the so called tensile instability is shown to be prevented by this enhanced sph method through accuracy increases in the gradient approximations

a hybrid particle mesh method for viscous incompressible multiphase flows a hybrid method to simulate unsteady multiphase flows in which a sharp interface separates incompressible fluids of different density and viscosity is described one phase is represented by moving particles and the other phase is defined on stationary mesh the flow field is discretized by a conservative finite volume approximation on the stationary mesh and the interface is automatically captured by the distribution of particles moving through the stationary mesh the effects of surface tension and wall adhesion are evaluated by the continuum surface force model the different phases are treated as one fluid with variable material properties advection of fluid properties such as density and viscosity is done by following the motion of the particles the method simplifies the calculation of interface interaction enables accurate modeling of two and three dimensional multiphase flows and does not impose any modeling restrictions on the dynamic evolutions of fluid interfaces having surface tension several two dimensional numerical simulations are given to illustrate the efficiency of the hybrid method

on a kinetic model for a simple market economy in this paper we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading we show that the kinetic model admits non trivial quasi stationary states with power law tails of pareto type in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution of wealth among individuals for this equation the stationary state can be easily derived and shows a pareto power law tail numerical results confirm the previous analysis

generalized potlatch and smoothing processes we consider simple generalizations of the potlatch and smoothing processes which were introduced in 8 and studied in 5 these generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition the original potlatch and smoothing processes do not exhibit phase transition our results show that for the generalized processes phase transition does not usually occur in one or two dimensions but usually does occur in higher dimensions upper and lower bounds for the relevant critical values are obtained as one application of our results we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d thus answering a question we raised in 3 this is done by comparing the contact process with an appropriate generalized smoothing process

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

simulating free surface flows with sph the sph smoothed particle hydrodynamics method is extended to deal with free surface incompressible flows the method is easy to use and examples will be given of its application to a breaking dam a bore the simulation of a wave maker and the propagation of waves towards a beach arbitrary moving boundaries can be included by modelling the boundaries by particles which repel the fluid particles the method is explicit and the time steps are therefore much shorter than required by other less flexible methods but it is robust and easy to program

kinetic models for the trading of goods in this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents the leading idea is to describe the trading of these goods by means of some fundamental rules in price theory in particular by using cobb douglas utility functions for the binary exchange and the edgeworth box for the description of the common exchange area in which utility is increasing for both agents this leads to a boltzmann type equation in which the post interaction variables depend in a nonlinear way from the pre interaction ones other models will be derived by suitably linearizing this boltzmann equation in presence of uncertainty in the exchanges it is shown that the solution to some of the linearized kinetic equations develop pareto tails where the pareto index depends on the ratio between the gain and the variance of the uncertainty in particular the result holds true for the solution of a drift diffusion equation of fokker planck type obtained from the linear boltzmann equation as the limit of quasi invariant trades

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

on the convergence of supercritical general c m j branching processes convergence in probability of malthus normed supercritical general branching processes i e crump mode jagers branching processes counted with a general characteristic are established provided the latter satisfies mild regularity conditions if the laplace transform of the reproduction point process evaluated in the malthusian parameter has a finite x log x moment convergence in probability of the empirical age distribution and more generally of the ratio of two differently counted versions of the process also follow

a natural element updated lagrangian strategy for free surface fluid dynamics we present a novel algorithm to simulate free surface fluid dynamics phenomena at low reynolds numbers in an updated lagrangian framework it is based on the use of one of the most recent meshless methods the so called natural element method free surface tracking is performed by employing a particular instance of shape constructors called a shapes this means that at each time step the geometry of the domain is extracted by employing a particular member of the finite set of shapes described by the nodal cloud the lagrangian framework allows us to integrate the inertial terms of the navier stokes equations by employing the method of characteristics which are precisely the nodal pathlines a theoretical description of the method is included together with some examples showing its performance

a modular particle continuum numerical method for hypersonic non equilibrium gas flows a modular particle continuum mpc numerical method for steady state flows is presented which solves the navier stokes equations in regions of near equilibrium and uses the direct simulation monte carlo dsmc method to simulate regions of non equilibrium gas flow existing state of the art dsmc and navier stokes solvers are coupled together using a novel modular implementation which requires only a limited number of additional hybrid functions hybrid functions are used to adaptively position particle continuum interfaces and update boundary conditions in each module at appropriate times the mpc method is validated for 2d flow over a cylinder at various hypersonic mach numbers where the global knudsen number is 0 01 for the cases considered the mpc method is verified to accurately reproduce dsmc flow field results as well as local particle velocity distributions up to 2 2 times faster than full dsmc simulations

the fixed points of the multivariate smoothing transform let mathbf t 1 mathbf t 2 ldots be a sequence of random d times d matrices with nonnegative entries and let q be a random vector with nonnegative entries consider random vectors x with nonnegative entries satisfying begin aligned x mathop limits mathcal l sum i ge 1 mathbf t i x i q end aligned where mathop limits mathcal l denotes equality of the corresponding laws x i i ge 1 are i i d copies of x and independent of q mathbf t 1 mathbf t 2 ldots for d 1 this equation known as fixed point equation of the smoothing transform has been intensively studied under assumptions similar to the one dimensional case we obtain a complete characterization of all solutions x to in the non critical case and existence results in the critical case

central limit theorem for a class of one dimensional kinetic equations we introduce a class of kinetic type equations on the real line which constitute extensions of the classical kac caricature the collisional gain operators are defined by smoothing transformations with rather general properties by establishing a connection to the central limit problem we are able to prove long time convergence of the equation s solutions toward a limit distribution for example we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law then the limit is a scale mixture of under some additional assumptions explicit exponential rates for the convergence to equilibrium in wasserstein metrics are calculated and strong convergence of the probability densities is shown

martingale convergence and the stopped branching random walk we discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines applying a method of lyons 1997 and lyons pemantle and peres 1995 concerning size biased branching trees we establish a relationship between stopping lines and certain stopping times consequently we develop conditions under which these supermartingales are also martingales further we prove a generalization of biggins martingale convergence theorem biggins 1977a within this context

the functional equation of the smoothing transform given a sequence t ti i 1 of non negative random variables a function f on the positive halfline can be transformed to e q i 1 f tti we study the fixed points of this transform within the class of decreasing functions by exploiting the intimate relationship with general branching processes a full description of the set of solutions is established without the moment conditions that figure in earlier studies since the class of functions under consideration contains all laplace transforms of probability distributions on 0 1 the results provide the full description of the set of solutions to the fixed point equation of the smoothing transform x d

on steady distributions of kinetic models of conservative economies we analyze the large time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years as specific examples we study models with fixed saving propensity introduced by chakraborti and chakrabarti eur phys j b 17 167 170 2000 as well as models involving both exchange between agents and speculative trading as considered by cordier et al j stat phys 120 253 277 2005 we derive a sufficient criterion under which a unique non trivial stationary state exists and provide criteria under which these steady states do or do not possess a pareto tail in particular we prove the absence of pareto tails in pointwise conservative models like the one in eur phys j b 17 167 170 2000 while models with speculative trades introduced in j stat phys 120 253 277 2005 develop fat tails if the market is risky enough the results are derived by a fourier based technique first developed for the maxwell boltzmann equation gabetta et al in j stat phys 81 901 934 1995 bisi et al in j stat phys 118 1 2 301 331 2005 pareschi and toscani in j stat phys 124 2 4 747 779 2006 and from a recursive relation which allows to calculate arbitrary moments of the stationary state

inelastically scattering particles and wealth distribution in an open economy using the analogy with inelastic granular gases we introduce a model for wealth exchange in society the dynamics is governed by a kinetic equation which allows for self similar solutions the scaling function has a power law tail the exponent being given by a transcendental equation in the limit of continuous trading a closed form of the wealth distribution is calculated analytically

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

convergence to stable laws for a class of multidimensional stochastic recursions we consider a markov chain x n n 0 infty on mathbb r d defined by the stochastic recursion xn mn xn 1 qn where qn mn are i i d random variables taking values in the affine group a mathbb r d mathbb r d rtimes rm gl mathbb r d assume that mn takes values in the group of similarities of mathbb r d and the markov chain has a unique stationary measure which has unbounded support we denote by mn the expansion coefficient of mn and we assume mathbb e m alpha 1 for some positive we show that the partial sums s n sum k 0 n x k properly normalized converge to a normal law 2 or to an infinitely divisible law which is stable in a natural sense 2 or of the tails of and of stationary measure for an associated markov operator 2 we extend the results to the situation where mn is a random generalized similarity

propagation of gevrey regularity for solutions of the boltzmann equation for maxwellian molecules we prove that gevrey regularity is propagated by the boltzmann equation with maxwellian molecules with or without angular cut off the proof relies on the wild expansion of the solution to the equation and on the characterization of gevrey regularity by the fourier transform

arbitrary lagrangian eulerian finite element analysis of free surface flow using a velocity vorticity formulation this paper describes the application of velocity vorticity formulation of the navier stokes equations for two dimensional free surface flow using an arbitrary lagrangian eulerian method the velocity poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain the boundary fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method the numerical model for the velocity vorticity formulation is validated for a square cavity flow at re 400 and 1000 the solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model then the proposed numerical model is used to obtain flow results for the following free surface flow cases i interaction between two opposite solitary waves ii seiche phenomenon in a rectangular reservoir and iii solitary wave through a submerged rectangular structure in a viscous fluid the efficiency of the present numerical model for numerical treatment of free surface flows is discussed furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view

free surface flows under impacting droplets a numerical method which fulfils the free surface boundary conditions and extrapolates the fluid velocity into empty grid cells outside the fluid region on a fixed cartesian grid system is presented the complex three dimensional vortex structures formed via surface vortex interaction and induction between vortices have been computed using the proposed technique implemented within a level set method for both vertical and oblique droplet impacts in incompressible fluids the present results have been validated through numerical tests which confirm zero tangential shear at the free surface and comparisons with experimental observations of cavity and vortex ring formation underneath the impact location in some cases transitions from a concentric vortex ring to a fully three dimensional vortex structure has been confirmed whilst the primary vortex ring is initiated at the highly curved contact surface between the droplet and receiving surface azimuthal instabilities are manifested in the shear layer around the cavity crater developing after the vertical impact resulting in axial counter rotating vorticity between the cavity and descending vortex ring underlying mechanisms which induce local deformation of the free surface creating a so called scar due to the sub surface vortices at the oblique impacts are also discussed

asymptotic properties of the inelastic kac model we discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of kac caricature of a maxwellian gas it is shown that global equilibria different from concentration are possible if the energy is not finite these equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction if the initial density is assumed of finite energy with higher moments bounded it is shown that the solution converges for large time to a profile with power law tails these tails are heavily dependent of the collision rule

hypam a hybrid continuum particle model for incompressible free surface flows three major issues associated with numerical simulations of complex free surface flows viz interface tracking fragmentation and large physical jumps are addressed by a new hybrid continuum particle model hypam the new model consists of three parts 1 the polygonal area mapping method q zhang p l f liu a new interface tracking method the polygonal area mapping method j comput phys 227 8 2008 4063 4088 2 a new algorithm that decomposes the interested water phase into a continuum zone a buffer zone and a particle zone based on material topology and graph theory 3 a passive response assumption in which the air phase is assumed to respond passively to the continuum part of the water phase the incompressible inviscid euler equations and the equations describing the free fall of rigid bodies are used as the governing equations for the continuum buffer zone and the particle zone respectively and separately a number of examples including water droplet impact solitary wave propagation and dam break problems are simulated for the illustration and validation of hypam it is shown that hypam is more accurate and versatile than a continuum based volume of fluid model one major contribution of this work is the single phase decomposition algorithm useful for many other hybrid formulations neglecting surface tension viscosity and particle interactions hypam is currently limited to mildly fragmented free surface flows with high reynolds and weber numbers

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

the joint distribution of elastic buckets in multiway search trees random search trees are studied when they grow under a general computer memory management scheme in a general scheme the space is released in buckets of certain predesignated sizes for a search tree with branch factor m the nodes may hold up to m 1 keys suppose the buckets of the memory management scheme that can hold less than m keys have key capacities c 1 ldots c p the search tree must then be implemented with multitype nodes of these capacities after n insertions let x n i be the number of buckets of type i i e of capacity c i 1 leq i leq p the multivariate structure of the tree is investigated for the vector bf x n x n 1 ldots x n p t the asymptotic mean and covariance matrix are determined under practical memory management schemes all variances and covariances experience a phase transition for 3 leq m leq 26 all variances and covariances are asymptotically linear in n for higher branch factors the variances and covariances become a superlinear but subquadratic function of n the joint distribution of bf x n is shown to be multivariate normal in a range of m while the tree is growing conversions between types are necessary a multivariate problem concerning these conversions with an asymptotic multivariate normal distribution is also studied the fixed bucket exact fit and buddy system allocation schemes will serve as illustrating examples

tail homogeneity of stationary measures for some multidimensional stochastic recursions we consider a stochastic recursion x n 1 m n 1 x n q n 1 n in mathbb n where q n m n are i i d random variables such that q n are translations m n are similarities of the euclidean space mathbb r d and x n in mathbb r d in the present paper we show that if the recursion has a unique stationary measure with unbounded support then the weak limit of properly dilated exists and defines a homogeneous tail measure the structure of is studied and the supports of and are compared in particular we obtain a product formula for

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

ripple a new model for incompressible flows with free surfaces a new free surface flow model ripple is summarised ripple obtains finite difference solutions for incompressible flow problems having significant surface tension forces at free surfaces of arbitrarily complex topology the key innovation is the continuum surface force model which represents surface tension as a localised volume force other features include a higher order momentum advection model a volume of fluid free surface treatment and an efficient two step projection solution method ripple s unique capabilities are illustrated with two example problems low gravity jet induced tank flow and the collision and coalescence of two cylindrical rods

decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model we quantify the long time behavior of solutions to the nonlinear boltzmann equation for spatially uniform freely cooling inelastic maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures existence uniqueness and precise estimates of overpopulated high energy tails of the self similar profile proved in ref 9 are revisited and derived from this new liapunov functional for general initial conditions the solutions of the boltzmann equation are then proved to converge with computable rate as t to the self similar solution in this distance which metrizes the weak convergence of measures moreover we can relate this fourier distance to the euclidean wasserstein distance or tanaka functional proving also its exponential convergence towards the homogeneous cooling states the findings are relevant in the understanding of the conjecture formulated by ernst and brito in refs 15 16 and complement and improve recent studies on the same problem of bobylev and cercignani 9 and bobylev cercignani and one of the authors 11

a second order boundary fitted projection method for free surface flow computations this paper describes a new approach to the high fidelity simulation of axisymmetric free surface flows a boundary fitted grid is coupled with a new projection method for the solution of the navier stokes equations with second order accuracy in space and time two variants of this new method are developed by adapting two existing algorithms suitable for prescribed velocity boundary conditions to the case of prescribed normal and tangential stresses at the free boundary a normal mode analysis for a fixed boundary problem confirms the second order accuracy of the algorithms the approach is validated by comparison with a rayleigh plesset solution for an oscillating spherical bubble with an analysis of shape oscillations and with existing results for the buoyant rise of a deforming bubble for reynolds numbers up to 200 and weber numbers up to 12 in addition to the simulation of axisymmetric free surface flows of intrinsic interest the present approach is suitable for the validation of genuinely three dimensional calculations

remark on invariant curves for planar mappings abstract bounded solutions for a functional equation were investigated by ng and zhang c t ng w zhang invariant curves for planar mapping j difference eqn appl 3 1997 147 168 in this paper we consider its solutions which may be unbounded we give not only existence and uniqueness of its solutions but also the convex and concave solutions moreover some examples are given

fourth order interface tracking in two dimensions via an improved polygonal area mapping method we present an improved pam ipam method as the first fourth order interface tracking method whose convergence rates are independent of c 1 derivative discontinuities of the interface as an improved version of the polygonal area mapping pam method q zhang and p l f liu j comput phys 227 2008 pp 4063 4088 the accuracy of the ipam method is achieved via i augmenting the abstract data structure of pam to faithfully represent multiple components of material regions within a single cell ii removing restrictive assumptions of pam iii adjusting the volume of represented cell material regions via polygon ear removal and iv maintaining a relation h l r hh alpha between the eulerian grid size h and the lagrangian length scale that measures the distance between adjacent interface markers unlike volume of fluid methods and level set methods merging and separation of the tracked material is not automatically handled by the ipam method instead flexible subgrid resolutions

wealth condensation in a simple model of economy abstract we introduce a simple model of economy where the time evolution is described by an equation capturing both exchange between individuals and random speculative trading in such a way that the fundamental symmetry of the economy under an arbitrary change of monetary units is insured we investigate a mean field limit of this equation and show that the distribution of wealth is of the pareto power law type the pareto behaviour of the tails of this distribution appears to be robust for finite range models as shown using both a mapping to the random directed polymer problem as well as numerical simulations in this context a phase transition between an economy dominated by a few individuals and a situation where the wealth is more evenly spread out is found an interesting outcome is that the distribution of wealth tends to be very broadly distributed when exchanges are limited either in amplitude or topologically favouring exchanges and less surprisingly increasing taxes seems to be an efficient way to reduce inequalities

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

on the self similar asymptotics for generalized nonlinear kinetic maxwell models maxwell models for nonlinear kinetic equations have many applications in physics dynamics of granular gases economics etc in the present manuscript we consider such models from a very general point of view including those with arbitrary polynomial non linearities and in any dimension space it is shown that the whole class of generalized maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual lipschitz conditions this property allows to describe in detail a behavior of solutions to the corresponding initial value problem in particular we prove in the most general case an existence of self similar solutions and study the convergence in the sense of probability measures of dynamically scaled solutions to the cauchy problem to those self similar solutions as time goes to infinity a new application of multi linear models to economics and social dynamics is discussed

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

on a family of unsplit advection algorithms for volume of fluid methods volume of fluid vof methods are widely used for the interface tracking problem yet rigorous analyses of them are rare this paper presents such an analysis for incompressible flows by combining the theories of ordinary differential equations differential geometry and boolean algebra based on the concept of donating region dr q zhang siam rev 55 2013 pp 443 461 the author classifies the fluxing particles of a fixed control volume into four categories and derives three analytical solutions for the advection equation of the color function one edgewise solution provides a unified view of dr based advection algorithms for vof methods while another cellwise solution serves as the theoretical foundation of the recent polygonal area mapping method the well known second order convergence rates of streamline based vof advection algorithms are proved rigorously potential deterioration of this second order convergence is also discussed for several subtle issues such as exact mass conservation a

a projection method for locally refined grids a numerical method for the solution of the two dimensional euler equatons for incompressible flow on locally refined grids is presented the method is a second order godunov projection method adapted from bell colella and glaz second order accuracy of the numerical method in time and space is established through numerical experiments the main contributions of this work concern the formulation and implementation of a projection for refined grids a discussion of the adjointness relation between gradient and divergence operators for a refined grid mac projection is presented and a refined grid approximate projection is developed an efficient multigrid method which exactly solves the projection is developed and a method for casting certain approximate projections as mac projections on refined grids is presented

a second order delay differential equation with multiple periodic solutions abstract the equation x t 2 x t bx t 1 where designates the greatest integer function can be described in brief by two amazing properties first for certain values of the coefficients some or all of its solutions are monotone although the corresponding homogeneous equation is clearly oscillatory second for a specific relation between and b there exist periodic solutions with different periods

analytic solutions of a functional equation for invariant curves abstract an iterative functional equation is deduced by c t ng and w zhang 1997 j differ equations appl 3 147 168 from the problem of invariant curves in this paper its analytic solutions are discussed by locally reducing the equation to another functional equation without iteration and by constructing solutions in uniformly convergent power series for the latter equation

kinetic equations modelling wealth redistribution a comparison of approaches kinetic equations modelling the redistribution of wealth in simple market economies is one of the major topics in the field of econophysics we present a unifying approach to the qualitative study for a large variety of such models which is based on a moment analysis in the related homogeneous boltzmann equation and on the use of suitable metrics for probability measures in consequence we are able to classify the most important feature of the steady wealth distribution namely the fatness of the pareto tail and the dynamical stability of the latter in terms of the model parameters our results apply e g to the market model with risky investments s cordier l pareschi and g toscani j stat phys 120 253 2005 and to the model with quenched saving propensities a chatterjee b k chakrabarti and s s manna physica a 335 155 2004 also we present results from numerical experiments that confirm the theoretical predictions

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions