homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

a field space based level set method for computing multi valued solutions to 1d euler poisson equations we present a field space based level set method for computing multi valued solutions to one dimensional euler poisson equations the system of these equations has many applications and in particular arises in semiclassical approximations of the schrodinger poisson equation the proposed approach involves an implicit eulerian formulation in an augmented space called field space which incorporates both velocity and electric fields into the configuration both velocity and electric fields are captured through common zeros of two level set functions which are governed by a field transport equation simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data the averaged density is then resolved by the integration of the obtained f against the dirac delta function of two level set functions in the field space moreover we prove that such obtained averaged density is simply a linear superposition of all multi valued densities and the averaged field quantities are weighted superposition of corresponding multi valued ones computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method

a bloch band based level set method for computing the semiclassical limit of schrodinger equations a novel bloch band based level set method is proposed for computing the semiclassical limit of schrodinger equations in periodic media for the underlying equation subject to a highly oscillatory initial data a hybrid of the wkb approximation and homogenization leads to the bloch eigenvalue problem and an associated hamilton jacobi system for the phase in each bloch band with the bloch eigenvalue be part of the hamiltonian we formulate a level set description to capture multi valued solutions to the band wkb system and then evaluate total homogenized density over a sample set of bands a superposition of band densities is established over all bands and solution branches when away from caustic points the numerical approach splits the solution process into several parts i initialize the level set function from the band decomposition of the initial data ii solve the bloch eigenvalue problem to compute bloch waves iii evolve the band level set equation to compute multi valued velocity and density on each bloch band iv evaluate the total position density over a sample set of bands using bloch waves and band densities obtained in steps ii and iii respectively numerical examples with different number of bands are shown to demonstrate the capacity of the method

numerical solution of the high frequency asymptotic expansion for the scalar wave equation new numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation the nonlinear partial differential equations defining the terms in the expansio

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

high order numerical methods to a type of delta function integrals we study second to fourth order numerical methods to a type of delta function integrals in one to three dimensions these delta function integrals arise from recent efficient level set methods for computing the multivalued solutions of nonlinear pdes we show that the natural quadrature approach with usual discrete delta functions and support size formulas to the two dimensional delta function integrals suffer from nonconvergence we then design high order numerical methods to this type of delta function integrals based on interpolation approach numerical examples are presented to verify the efficiency and accuracy of our methods

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

a quadrature based third order moment method for dilute gas particle flows dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables a useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function closure of the moment equations is challenging for flows away from the equilibrium maxwellian limit in this work a quadrature based third order moment closure is derived that can be applied to gas particle flows at any knudsen number a key component of quadrature based closures is the moment inversion algorithm used to find the weights and abscissas a robust inversion procedure is proposed for moments up to third order and tested for three example applications riemann shock problem impinging jets and vertical channel flow extension of the moment inversion algorithm to fifth or higher order is possible but left to future work the spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient diffusion model is not used to close the fluxes because the quadrature based moment method employs the moment transport equations directly instead of a discretized form of the boltzmann equation the mass momentum and energy are conserved for arbitrary knudsen number including the euler limit while developed here for dilute gas particle flows quadrature based moment methods can in principle be applied to any application that can be modeled by a kinetic equation e g thermal and non isothermal flows currently treated using lattice boltzmann methods and examples are given from the literature

computing multivalued physical observables for the semiclassical limit of the schrodinger equation we develop a level set method for the computation of multivalued physical observables density velocity etc for the semiclassical limit of the schrodinger equation this method uses an eulerian formulation and applies directly to arbitrary number of space dimensions the main idea is to evolve the density near an n dimensional manifold that is identified as the common zeros of n level set functions in phase space these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity f by solving again the liouville equation near the obtained zero level set but with initial density as initial data the multivalued density and higher moments are thus resolved by integrating f along the n dimensional manifold in the phase directions we show that this is equivalent to using the wigner approach but decomposing the velocity from the density each of which evolves by the same liouville equation the main advantages of this approach in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these advantages allow us to compute for the first time all physical observables for multidimensional problems in an eulerian framework

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

fast phase space computation of multiple arrivals we present a fast general computational technique for computing the phase space solution of static hamilton jacobi equations starting with the liouville formulation of the characteristic equations we derive escape equations which are static time independent eulerian pdes they represent all arrivals to the given boundary from all possible starting configurations the solution is numerically constructed through a one pass formulation building on ideas from semi lagrangian methods dijkstra like methods for the eikonal equation and ordered upwind methods to compute all possible trajectories corresponding to all possible boundary conditions the technique is of computational order o n log n where n is the total number of points in the computational phase space domain any particular set of boundary conditions then is extracted through rapid post processing suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance as an application we apply the technique to the problem of computing first multiple and most energetic arrivals to the eikonal equation

using k branch entropy solutions for multivalued geometric optics computations this paper is devoted to a numerical simulation of the classical wkb system arising in geometric optics expansions it contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous we address the problem of treating it in such a way that superimposed signals can be reproduced by means of the kinetic formulation of multibranch solutions originally due to brenier and corrias some existence and uniqueness results are given together with computational test cases of increasing difficulty displaying up to five multivaluations

big ray tracing different traveltimes are identified as viscosity solutions of the eikonal equation on locally selected domains called big rays big rays are constructed by an ad hoc algorithm based on a coarse ray tracing viscosity solutions in the big rays are computed using an upwind finite difference scheme this hybrid method allows an easy and fast computation of multivalued traveltime fields

computational high frequency waves through curved interfaces via the liouville equation and geometric theory of diffraction we construct a class of numerical schemes for the liouville equation of geometric optics coupled with the geometric theory of diffractions to simulate the high frequency linear waves with a discontinuous index of refraction in this work s jin x wen a hamiltonian preserving scheme for the liouville equation of geometric optics with partial transmissions and reflections siam j numer anal 44 2006 1801 1828 a hamiltonian preserving scheme for the liouville equation was constructed to capture partial transmissions and reflections at the interfaces this scheme is extended by incorporating diffraction terms derived from geometric theory of diffraction into the numerical flux in order to capture diffraction at the interface we give such a scheme for curved interfaces this scheme is proved to be positive under a suitable time step constraint numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation

multi phase computations in geometrical optics in this work we propose a new set of partial differential equations pdes which can be seen as a generalization of the classical eikonal and transport equations to allow for solutions with multiple phases the traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited in particular it does not include solutions with multiple phases corresponding to crossing waves our objective has been to generalize these equations to accommodate solutions containing more than one phase the new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations however they also incorporate a finite superposition principle the maximum allowed number of intersecting waves in the solution can be chosen arbitrarily but a higher number means that a larger system of pdes must be solved the pdes form a hyperbolic system of conservation laws with source terms although the equations are only weakly hyperbolic and thus not well posed in the strong sense several examples show the viability of solving the equations numerically the technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments

existence of classical solutions to nonautonomous nonlocal parabolic problems abstract we obtain a new existence theorem for classical solutions to nonautonomous equations with nonlocal initial conditions u t a t u t f t u t t s t u s g u u 0 in a banach space x where t s 0 f g are given x valued functions and a t is a sectorial operator not necessarily densely defined in x for each t 0 t both banach s contraction principle and schauder s fixed point theorem as well as the theory of evolution families and interpolation spaces are employed in our approach a concrete example is shown to illustrate the existence theorem

direct computation of multivalued phase space solutions for hamilton jacobi equations a method is presented for the exact and complete resolution of hamiltonian systems only using the corresponding hamilton jacobi equation we show that it possible to get the ode solution from the corresponding pde

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials we propose a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials the discontinuities in the potential corresponds to potential barriers at which incoming waves can be partially transmitted and reflected previously such a problem was handled by jin and wen using the liouville equation which arises as the semiclassical limit of the schrodinger equation with an interface condition to account for partial transmissions and reflections s jin x wen siam j num anal 44 2006 1801 1828 however the initial data are dirac delta functions which are difficult to approximate numerically with a high accuracy in this paper we extend the level set method introduced in s jin h liu s osher r tsai j comp phys 210 2005 497 518 for this problem instead of directly discretizing the delta functions our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow and hence can be solved much more easily using conventional high order discretization schemes two ideas are introduced here 1 the solutions of the problems involving partial transmissions and partial reflections are decomposed into a finite sum of solutions solving problems involving only complete transmissions and those involving only complete reflections for problems involving only complete transmission or complete reflection the method of jlot applies and is used in our simulations 2 a reinitialization technique is introduced so that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems this is implemented by rewriting the sum of several delta functions as one delta function with a suitable weight which can be easily implemented numerically we carry out numerical experiments in both one and two space dimensions to verify this new algorithm

a level set approach for computing discontinuous solutions of hamilton jacobi equations we introduce two types of finite difference methods to compute the l solution and the proper viscosity solution recently proposed by the second author for semi discontinuous solutions to a class of hamilton jacobi equations by regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher we can treat the corresponding level set equation using the viscosity theory introduced by crandall and lions however we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function we demonstrate our lax friedrichs type numerical methods for computing the l solution using its original level set formulation in addition we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of hj equations that includes conservation laws with this singular viscosity our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws these numerical methods are generalized to higher order accuracy using weighted eno local lax friedrichs methods as developed recently by jiang and peng we verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent hokkaido university preprint finally since the solution of scalar conservation law equations can be constructed using existing numerical techniques we use it to verify that our numerical solution approximates the entropy solution

high frequency wave propagation by the segment projection method geometrical optics is a standard technique used for the approximation of high frequency wave propagation computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics these new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves in this paper we introduce a new partial differential technique based on the segment projection method in phase space the superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics the computational complexity is of the same order as that of ray tracing the new algorithm is described and a number of computational examples are given including a simulation of waveguides

an accurate spectral discontinuous finite element formulation of a phase space based level set approach to geometrical optics in this paper we introduce a new numerical procedure for simulations in geometrical optics that based on the recent development of eulerian phase space formulations of the model can deliver very accurate uniformly resolved solutions which can be made to converge with arbitrarily high orders in general geometrical configurations following previous treatments the scheme is based on the evolution of a wavefront in phase space defined as the intersection of level sets satisfying the relevant liouville equation in contrast with previous work however our numerical approximation is specifically designed i to take full advantage of the smoothness of solutions ii to facilitate the treatment of scattering obstacles all while retaining high order convergence characteristics indeed to incorporate the full regularity of solutions that results from the unfolding of singularities our method is based on their spectral representation to enable a simple high order treatment of scattering boundaries on the other hand we resort to a discontinuous galerkin finite element method for the solution of the resulting system of equations the procedure is complemented with the use of a recently derived strong stability preserving runge kutta ssp rk scheme for the time integration that as we demonstrate allows for overall approximations that are rapidly convergent

simple bounds on fluctuations and uncertainty relations for first passage times of counting observables recent large deviation results have provided general lower bounds for the fluctuations of time integrated currents in the steady state of stochastic systems a corollary are so called thermodynamic uncertainty relations connecting precision of estimation to average dissipation here we consider this problem but for counting observables i e trajectory observables which in contrast to currents are non negative and nondecreasing in time and possibly symmetric under time reversal in the steady state their fluctuations to all orders are bound from below by a conway maxwell poisson distribution dependent only on the averages of the observable and of the dynamical activity we show how to obtain the corresponding bounds for first passage times times when a certain value of the counting variable is first reached and their uncertainty relations just like entropy production does for currents dynamical activity controls the bounds on fluctuations of counting observables

a level set approach for dilute non collisional fluid particle flows gas particle and other dispersed phase flows can be described by a kinetic equation containing terms for spatial transport acceleration and particle processes such as evaporation or collisions however computing the dispersed velocity is a challenging task due to the large number of independent variables a level set approach for computing dilute non collisional fluid particle flows is presented we will consider the sprays governed by the williams kinetic equation subject to initial distributions away from equilibrium of the form i 1 n i x u i x the dispersed velocity is described as the zero level set of a smooth function which satisfies a transport equation this together with the density weight recovers the particle distribution at any time moments of any desired order can be evaluated by a quadrature formula involving the level set function and the density weight it is shown that the method can successfully handle highly non equilibrium flows e g impinging particle jets jet crossing particle rebound off walls finite stokes number flows