automorphisms of the nottingham group abstract let k be a finite field of characteristic p 5 the nottingham group n over k is the group of normalised automorphisms of the local field k t in this paper we determine the automorphism group of n in particular we show that every automorphism of the nottingham group is standard we also give a complete description of the conjugacy classes of elements of order p in n

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

highly localized solutions of the wave equation simple explicit solutions of the linear wave equation in three dimensions are presented which describe wave packets exponentially localized near a point moving with the wave speed for large values of a certain free parameter these new solutions are localized in gaussian manner with respect to longitudinal and transverse variables and time this agrees with considerations by babich ulin and ralston who have presented an asymptotic description of solutions exhibiting such local behavior global estimates and large time asymptotics of these solutions are given

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

path space measure for the 3 1 dimensional dirac equation in momentum space nonstandard analysis is used to construct a measure over paths for the path integral solution to the dirac equation in 3 1 dimension paths are considered in momentum space because the green function in the configuration space contains a derivative of function which keeps us from assigning a measure over paths the solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure but also as a standard path integral with respect to a standard measure extracted from the nonstandard one the result is an extension of gaveau s work b gaveau j funct anal 58 310 319 1984

fast gaussian wavepacket transforms and gaussian beams for the schrodinger equation this paper introduces a wavepacket transform based gaussian beam method for solving the schrodinger equation we focus on addressing two computational issues of the gaussian beam method how to generate a gaussian beam representation for general initial conditions and how to perform long time propagation for any finite period of time to address the first question we introduce fast gaussian wavepacket transforms and develop on top of them an efficient initialization algorithm for general initial conditions based on this new initialization algorithm we address the second question by reinitializing the beam representation when the beams become too wide numerical examples in one two and three dimensions demonstrate the efficiency and accuracy of the proposed algorithms the methodology can be readily generalized to deal with other semi classical quantum mechanical problems

the nonexistence of the path space measure for the dirac equation in four space time dimensions it is proved that the path space measure for the dirac equation in four space time dimensions does not exist the origin of the nonexistence of the measure turns out to be the dependence of the solution on the first derivative of the initial condition

the space of state vectors a hyperfinite approach we present a version of the formalism of dirac based on nonstandard analysis allowing us to deal with state vectors and operators using the resources of finite dimensional linear algebra the space of state vectors is a nonstandard hilbert space with hyperfinite dimension which includes all square integrable functions together with vectors representing states of definite position or momentum every vector is normalizable even when its norm is infinite observables are represented by hermitian operators which are always hyper bounded and defined on the whole space the connection with the standard theory is established by postulating the existence of hyper observables and nonstandard states each observable in the usual sense appears as a kind of standard scale approximation of some hyper observable we show that the probabilistic predictions are consistent with those of the standard theory consistency extends to time evolution in the sense that if an initial nonstandard state is near standard then the state after a finite time shall be infinitely near the standard state obtained through the schrodinger equation

bloch decomposition based gaussian beam method for the schrodinger equation with periodic potentials the linear schrodinger equation with periodic potentials is an important model in solid state physics the most efficient direct simulation using a bloch decomposition based time splitting spectral method 18 requires the mesh size to be o e where e is the scaled semiclassical parameter in this paper we generalize the gaussian beam method introduced in jin et al 23 to solve this problem asymptotically we combine the technique of bloch decomposition and the eulerian gaussian beam method to arrive at an eulerian computational method that requires mesh size of o e the accuracy of this method is demonstrated via several numerical examples

gaussian beam methods for the schrodinger equation with discontinuous potentials we propose eulerian and lagrangian gaussian beam methods for the schrodinger equation with discontinuous potentials at the quantum barriers where the potential is discontinuous we derive suitable interface conditions to account for quantum scattering information these scattering interface conditions are then built into the numerical fluxes in the eulerian level set formulation of the gaussian beam methods and are also used in the lagrangian formulation including an interface condition for the hessian matrix we carry both 1d and 2d numerical examples to verify the accuracy of the method

embedding properties of hereditarily just infinite profinite wreath products abstract we study infinitely iterated wreath products of finite permutation groups w r t product actions in particular we prove that for every non empty class of finite simple groups x there exists a finitely generated hereditarily just infinite profinite group w with composition factors in x such that any countably based profinite group with composition factors in x can be embedded into w additionally we investigate when infinitely iterated wreath products of finite simple groups w r t product actions are co hopfian or non co hopfian

pro p groups with few normal subgroups abstract motivated by the study of pro p groups with finite coclass we consider the class of pro p groups with few normal subgroups this is not a well defined class and we offer several different definitions and study the connections between them furthermore we propose a definition of periodicity for pro p groups thus providing a general framework for some periodic patterns that have already been observed in the existing literature we then focus on examples and show that strikingly all the interesting examples not only have few normal subgroups but in addition have periodicity in the lattice of normal subgroups

path space measures for dirac and schrodinger equations nonstandard analytical approach a nonstandard path space measure is constructed to justify the path integral formula for the dirac equation in two dimensional space time a standard measure as well as a standard path integral is obtained from it we also show that even for the schrodinger equation for which there is no standard measure appropriate for a path integral there exists a nonstandard measure to define a path integral whose standard part agrees with the ordinary path integral as defined by a limit from time slice approximant

fast multiscale gaussian beam methods for wave equations in bounded convex domains motivated by fast multiscale gaussian wavepacket transforms and multiscale gaussian beam methods which were originally designed for pure initial value problems of wave equations we develop fast multiscale gaussian beam methods for initial boundary value problems of wave equations in bounded convex domains in the high frequency regime to compute the wave propagation in bounded convex domains we have to take into account reflecting multiscale gaussian beams which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation to propagate multiscale beams efficiently we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved and accordingly we propose an effective indicator to identify significant beams we also prove that the resulting multiscale gaussian beam methods converge asymptotically numerical examples demonstrate the accuracy and efficiency of the method

gaussian beam decomposition of high frequency wave fields in this paper we present a method of decomposing a highly oscillatory wave field into a sparse superposition of gaussian beams the goal is to extract the necessary parameters for a gaussian beam superposition from this wave field so that further evolution of the high frequency waves can be computed by the method of gaussian beams the methodology is described for r d with numerical examples for d 2 in the first example a field generated by an interface reflection of gaussian beams is decomposed into a superposition of gaussian beams the beam parameters are reconstructed to a very high accuracy the data in the second example is not a superposition of a finite number of gaussian beams the wave field to be approximated is generated by a finite difference method for a geometry with two slits the accuracy in the decomposition increases monotonically with the number of beams

abstract commensurators of profinite groups in this paper we initiate a systematic study of the abstract com mensurators of profinite groups the abstract commensurator of a profinite group g is a group comm g which depends only on the commensurability class of g we study various properties of comm g in particular we find two natural ways to turn it into a topological group we also use comm g to study topological groups which contain g as an open subgroup all such groups are totally disconnected and locally compact for instance we con struct a topologically simple group which contains the pro 2 completion of the grigorchuk group as an open subgroup on the other hand we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups several celebrated rigidity theorems like pink s analogue of mostow s strong rigidity theorem for simple algebraic groups defined over local fields and the neukirch uchida theorem can be re formulated as structure theorems for the commensurators of certain profinite groups

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

eulerian gaussian beams for high frequency wave propagation we design an eulerian gaussian beam summation method for solving helmholtz equations in the high frequency regime the traditional gaussian beam summation method is based on lagrangian ray tracing and local ray centered coordinates we propose a new eulerian formulation of gaussian beam theory which adopts global cartesian coordinates level sets and liouville equations yielding uniformly distributed eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources the time harmonic wavefield can be constructed by summing up gaussian beams with ingredients provided by the new eulerian formulation the conventional gaussian beam summation method can be derived from the proposed method there are three advantages of the new method 1 we have uniform resolution of ray distribution 2 we can obtain wavefields excited at different sources by varying only source locations in the summation formula 3 we can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula numerical experiments indicate that the gaussian beam summation method yields accurate asymptotic wavefields even at caustics the new method may be used for seismic modeling and migration

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

high order numerical methods to two dimensional delta function integrals in level set methods in this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in x wen high order numerical methods to a type of delta function integrals j comput phys 226 2007 1952 1967 we establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods

new just infinite pro p groups of finite width and subgroups of the nottingham group we use the action of the nottingham group on the completion of its lie algebra to construct new examples of just infinite pro p groups of finite width whose graded lie algebras with respect to the lower central series are isomorphic to sl2 fp tfp t the groups constructed are non linear and thus answer the question posed in comm algebra 30 2002 1293 the proof of nonlinearity uses a new description of the centralizers of elements of order p in the nottingham group and the concept of hausdorff dimension

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

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

high order numerical quadratures to one dimensional delta function integrals we study high order numerical quadratures to one dimensional delta function integrals in this paper this is motivated by the fact that traditional numerical quadratures give only first order accuracy in general we provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures discrete delta functions and support size formulas satisfying these criteria are designed numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures

on the commensurator of the nottingham group let p 5 be a prime number we prove that the abstract commensurator of the nottingham group n f p coincides with its automorphism group which is known to be a finite extension of n f p as a corollary we deduce that the nottingham group cannot be embedded as an open subgroup of a topologically simple group

error estimates for gaussian beam superpositions gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations concentrated on a single curve through the physical domain they can also be extended to some dispersive wave equations such as the schrodinger equation superpositions of gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve this work is concerned with the accuracy of gaussian beam superpositions in terms of the wavelength epsilon we present a systematic construction of gaussian beam superpositions for all strictly hyperbolic and schrodinger equations subject to highly oscillatory initial data of the form ae i phi epsilon through a careful estimate of an oscillatory integral operator we prove that the k th order gaussian beam superposition converges to the original wave field at a rate proportional to epsilon k 2 in the appropriate norm dictated by the well posedness estimate in particular we prove that the gaussian beam superposition converges at this rate for the acoustic wave equation in the standard epsilon scaled energy norm and for the schrodinger equation in the l 2 norm the obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics we present a numerical study of convergence for the constant coefficient acoustic wave equation in r 2 to analyze the sharpness of the theoretical results

index subgroups of the nottingham group abstract it is known that every countably based pro p group embeds as a closed subgroup into the nottingham group j j f p at the same time the subgroup structure of j remains rather mysterious in this paper we introduce the class of so called index subgroups of j and study their fundamental properties two main applications are given i we prove that a certain infinite subclass of index subgroups provides new examples of hereditarily just infinite pro p groups of finite width ii we determine which hausdorff dimensions occur for index subgroups and subsequently show that the hausdorff spectrum of j begins with a proper interval

prestack gaussian beam depth migration kirchhoff migration is the most popular method of three dimensional prestack depth migration because of its flexibility and efficiency its effectiveness can become limited however when complex velocity structure causes multipathing of seismic energy an alternative is gaussian beam migration which is an extension of kirchhoff migration that overcomes many of the problems caused by multipathing unlike first arrival and most energetic arrival methods which retain only one traveltime this alternative method retains most arrivals by the superposition of gaussian beams this paper presents a prestack gaussian beam migration method that operates on common offset gathers the method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points gaussian beam migration of the two dimensional marmousi test data set demonstrates the method s effectiveness for structural imaging in a case w

a nonstandard representation of feynman s path integrals a nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the cauchy problem for the dirac equation in 1 1 dimensional space time nonstandard analysis makes the mathematical concepts elementary consequently the procedures to prove theorems are considerably simplified a difference scheme with infinitesimal spacing is available in determining the probability distribution over the path space and this method is also available for the heat equation and for the free schrodinger equation

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

coarse grained spin density functional theory infinite volume limit via the hyperfinite coarse grained spin density functional theory sdft is a version of sdft which works with number spin densities specified to a limited resolution averages over cells of a regular spatial partition and external potentials constant on the cells this coarse grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation problems of existence uniqueness and regularity of representing potentials in the coarse grained sdft setting are here studied using techniques of robinsonian nonstandard analysis every density which is nowhere spin saturated is v representable and the set of representing potentials is the functional derivative in an appropriate generalized sense of the lieb internal energy functional quasi continuity and closure properties of the set valued representing potentials map are also established the extent of possible non uniqueness is similar to that found in non rigorous studies

representation formulas of the cauchy problem for hyperbolic systems generalizing dirac system on considere l equation de dirac dans un potentiel electromagnetique externe qui ne depend pas du temps et on donne une version rigoureuse de l integrale de parcours de feynman on considere l energie cinetique comme une perturbation du potentiel et on effectue une transformee de fourier sur des variables d espace de facon que le potentiel electromagnetique devienne un operateur de convolution a valeur matricielle et l energie cinetique devient une multiplication par une fonction a valeur matricielle de p on obtient la solution du probleme de cauchy comme une esperance mathematique sur un processus de poisson construit avec le potentiel

gaussian wave packets in inhomogeneous media with curved interfaces time dependent particle like pulses are considered as asymptotic solutions of the classical wave equation the wave packets are localized in space with gaussian envelopes the pulse centres propagate along the rays of the wave equation and the envelope parameters satisfy evolution equations very similar to the ray equations for time harmonic disturb ances however the present theory contains an extra degree of freedom not found in the time harmonic theory explicit results are presented for media with constant velocity gradients and interesting new phenomena are identified for example a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction maintains its initial spatial orientation even as the propagation direction rotates the reflection and transmission of a pulse incident upon an interface are also discussed the various theoretical results are illustrated by numerical simulations this method of solution could be very useful for fast forward modelling in large scale structures it is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics

gaussian beam decomposition of high frequency wave fields using expectation maximization a new numerical method for approximating highly oscillatory wave fields as a superposition of gaussian beams is presented the method estimates the number of beams and their parameters automatically this is achieved by an expectation maximization algorithm that fits real positive gaussians to the energy of the highly oscillatory wave fields and its fourier transform beam parameters are further refined by an optimization procedure that minimizes the difference between the gaussian beam superposition and the highly oscillatory wave field in the energy norm

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not numerical test examples are presented to help interpret the theory

feynman kac formula associated with a system of partial differential operators abstract the theory of the feynman path integral by albeverio and hoegh krohn is extended to a wider class of operators which involves the dirac operator and the schrodinger operator the fundamental solutions are obtained in the form of the feynman kac formula in this context the convergence of a product integral of operators is proved

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

eulerian gaussian beams for schrodinger equations in the semi classical regime we propose gaussian beam based eulerian methods to compute semi classical solutions of the schrodinger equation traditional gaussian beam type methods for the schrodinger equation are based on the lagrangian ray tracing based on the first eulerian gaussian beam framework proposed in leung et al s leung j qian r burridge eulerian gaussian beams for high frequency wave propagation geophysics 72 2007 sm61 sm76 we develop a new eulerian gaussian beam method which uses global cartesian coordinates level set based implicit representation and liouville equations the resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously to obtain semi classical solutions to the schrodinger equation with different initial wave functions we only need to slightly modify the summation formula this yields a very efficient method for computing semi classical solutions to the schrodinger equation for instance in the one dimensional case the proposed algorithm requires only o snm2 operations to compute s different solutions with s different initial wave functions under the influence of the same potential where n o 1 is the planck constant and m n is the number of computed beams which depends on weakly numerical experiments indicate that this eulerian gaussian beam approach yields accurate semi classical solutions even at caustics

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy

high order numerical methods to three dimensional delta function integrals in level set methods in this paper we propose a class of high order numerical methods to delta function integrals appearing in level set methods in the three dimensional case by extending the idea for designing high order methods to two dimensional delta function integrals in x wen j comput phys 228 2009 pp 4273 4290 the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the three dimensional delta function integral can be rewritten as a two dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral a key idea in designing high order methods in this paper is that the high order accuracy of the methods is not ensured in each mesh cell and the methods are designed in a consistent way to foster error cancelations in mesh cells where high order accuracy is not ensured in the individual cell this issue is essentially related to the construction of integral area for the two dimensional ordinary integral in each mesh cell which is achieved by considering the intersection points between the zero level set and the edges or sides of the mesh cell the mesh cell restrictions of the three dimensional delta function integral are then approximated by applying standard two dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals numerical examples are presented showing that our methods in this paper achieve or exceed the expected second to fourth order accuracy and demonstrating the stability of shifting mesh for our methods and the advantage of using our high order numerical methods