approximations of temporal rainfall from a multidimensional model the feasibility of representing the temporal structure of a multidimensional rainfall process with simpler stochastic models and a study of the effect of parameter robustness on the time scale is investigated here via performing controlled numerical experiments a multidimensional representation for precipitation given in the theory recently proposed by e waymire et al 1984 is used for simulating rainfall in space and time the model produces moving storms with realistic mesoscale meteorological features e g clustering birth and death of cells cell intensity attenuation in time and space etc two year traces of rainfall intensities at fixed gage stations were generated at intervals of 0 1 hours for three climates these traces are then aggregated at different time scales ranging from 1 to 24 hours first and second order statistics are evaluated from the above series at each aggregation level and they are used for estimating the parameters of three one dimensional models of temporal rainfall at a point 1 poisson model with independent marks 2 rectangular pulses with independent intensity and duration and 3 neyman scott with independent marks only at very high levels of aggregation the compound poisson model was able to reproduce the statistical structure of the simulated traces the rectangular pulses model was found to have parameters which vary significantly with the time scale of aggregation and moreover it leads to large distortions in the mean depth and duration of storm events the neyman scott scheme proved to be the most stable model throughout the different time scales and it also reproduced quite well the average storm characteristics at the event level none of these three models was able to reproduce in a satisfactory manner the simulated extreme value distribution of the multidimensional model

the mathematical structure of rainfall representations 3 some applications of the point process theory to rainfall processes part 3 constitutes the final part of three parts devoted to the mathematical structure of rainfall the objective is to illustrate the scope of the tools developed in part 2 in the mathematical description of rainfall and rainfall driven processes a general overview of the three part series is given in part 1

velocity boundary condition at solid walls in rarefied gas calculations maxwell s famous slip boundary condition is often misapplied in current rarefied gas flow calculations e g in hypersonics microfluidics for simulations of gas flows over curved or moving surfaces this means crucial physics can be lost we give examples of such cases we also propose a higher order boundary condition based on maxwell s general equation and the constitutive relations derived by burnett unlike many other higher order slip conditions these are applicable to any form of surface geometry it is shown that these maxwell burnett boundary conditions are in reasonable agreement with the limited experimental data available for poiseuille flow and can also predict sone s thermal stress slip flow a phenomenon which cannot be captured by conventional slip boundary conditions

a multidimensional gas kinetic bgk scheme for hypersonic viscous flow this paper concerns the development and application of a gas kinetic scheme based on the bhatnagar gross krook bgk model for the navier stokes equations in the study of hypersonic viscous flow firstly we extend the gas kinetic navier stokes solver k xu a gas kinetic bgk scheme for the navier stokes equations and its connection with artificial dissipation and godunov method j comput phys 171 2001 289 335 by implementing a multidimensional particle propagation mechanism in the flux evaluation where the gradients of flow variables in both normal and tangential directions of a cell interface are explicitly included with the construction of a time averaged flux function an implicit bgk scheme with lu sgs method is constructed the main purpose of the current research is to pave the way to extend the current approach directly to the flow computation with unstructured mesh where the flow gradients in both parallel and perpendicular directions around a cell interface can be explicitly taken into account in a viscous flow computation in the numerical parts we concentrate on the computation of heat flux in laminar hypersonic viscous flows where complicated flow phenomena i e shock boundary layer interaction flow separation and viscous inviscid interaction will be encountered the cases studied include the type iv shock shock interaction around a circular cylinder and hypersonic flow passing through a double cone geometry in the hypersonic viscous flow in comparison with the capturing of velocity and pressure fields the accurate computation of stress and heat flux bears large difficulties in all cases studied here the heat fluxes obtained across body surfaces have good agreement with the experimental measurements

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

the kinetic scheme for the full burnett equations the present paper concerns two aspects for the burnett equations first we are going to theoretically show the consistency between the traditional chapman enskog expansion and the successive approximation for the bgk equation up to the super burnett order second we will design a numerical scheme to efficiently solve the burnett equations the current approach is an improvement of the bgk burnett scheme k xu phys fluids 15 2003 2077 where the locally constant collision frequency is considered based on the burnett distribution function a high order time accurate numerical flux is derived by different approaches the resulting scheme is tested in the problem of normal shock wave and that of force driven poiseuille flow

on the largest bell violation attainable by a quantum state abstract we study the projective tensor norm as a measure of the largest bell violation of a quantum state in order to do this we consider a truncated version of a well known sdp relaxation for the quantum value of a two prover one round game one which has extra restrictions on the dimension of the sdp solutions our main result provides a quite accurate upper bound for the distance between the classical value of a bell inequality and the corresponding value of the relaxation along the way we give a simple proof that the best complementation constant of l 2 n in l 1 l is of order ln n as a direct consequence we show that we cannot remove a logarithmic factor when we are computing the largest bell violation attainable by the maximally entangled state

lattice boltzmann method and gas kinetic bgk scheme in the low mach number viscous flow simulations both lattice boltzmann method lbm and the gas kinetic bgk scheme are based on the numerical discretization of the boltzmann equation with collisional models such as the bhatnagar gross krook bgk model lbm tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process on the other hand the gas kinetic bgk scheme is a finite volume scheme where the time dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces currently lbm is mainly used for low mach number nearly incompressible flow simulation for the gas kinetic scheme the application is focusing on the high speed compressible flows in this paper we are going to compare both schemes in the isothermal low mach number flow simulations the methodology for developing both schemes will be clarified through the introduction of operator splitting boltzmann model and operator averaging boltzmann model from the operator splitting boltzmann model the error rooted in many kinetic schemes which are based on the decoupling of particle transport and collision can be easily understood as to the test case we choose to use the 2d cavity flow since it is one of the most extensively studied cases detailed simulation results with different reynolds numbers as well as the benchmark solutions are presented

inverted velocity profile in the cylindrical couette flow of a rarefied gas the cylindrical couette flow of a rarefied gas is investigated under the diffuse specular reflection condition of maxwell s type on the cylinders in the case where the inner cylinder is rotating whereas the outer cylinder is at rest the inverted velocity profile for small accommodation coefficients pointed out by tibbs baras and garcia phys rev e 56 2282 1997 on the basis of a monte carlo simulation is investigated extensively by means of a systematic asymptotic analysis for small knudsen numbers as well as the direct numerical analysis of the boltzmann equation and the parameter range in which the phenomenon appears is clarified

lower upper symmetric gauss seidel method for the euler and navier stokes equations on developpe un schema de relaxation multigrille application a des ecoulements transsoniques

a third order gas kinetic cpr method for the euler and navier stokes equations on triangular meshes abstract a third order accurate gas kinetic scheme based on the correction procedure via reconstruction cpr framework is developed for the euler and navier stokes equations on triangular meshes the scheme combines the accuracy and efficiency of the cpr formulation with the multidimensional characteristics and robustness of the gas kinetic flux solver comparing with high order finite volume gas kinetic methods the current scheme is more compact and efficient by avoiding wide stencils on unstructured meshes unlike the traditional cpr method where the inviscid and viscous terms are treated differently the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly through the kinetic evolution model in addition the present scheme adopts a fully coupled spatial and temporal gas distribution function for the flux evaluation achieving high order accuracy in both space and time within a single step numerical tests with a wide range of flow problems from nearly incompressible to supersonic flows with strong shocks for both inviscid and viscous problems demonstrate the high accuracy and efficiency of the present scheme

the mathematical structure of rainfall representations 2 a review of the theory of point processes part 2 constitutes the second of three parts devoted to the mathematical structure of rainfall an expository yet current treatment of those features of point process theory which are essential to descriptions of the hydrologic processes is developed a general overview of this three part series is given in part 1

on the multidimensional gas kinetic bgk scheme this paper concerns the performance difference among the multidimensional md gas kinetic bgk scheme the corresponding quasi one dimensional q1d extension and the directional splitting ds scheme with kinetic boundary conditions the md scheme includes tangential slopes in the flux calculation which are absent in a q1d or ds method in spite of taking more computational time the md scheme is found to be able to capture the characteristics of rarefied flow having a boundary with curvature or a nonuniform temperature such as the inverted velocity distribution in rarefied cylindrical couette flow and the weak flow field induced by the temperature gradient of a body where the q1d fails the md scheme can also yield clearly better results in high speed microchannel flow and power law fluid flow between concentric rotating cylinders when compared with direct simulation monte carlo studies and analytic solutions in the low reynolds number flow around a naca0012 airfoil case the md scheme and the q1d method predict wall heat flux distribution with obvious difference the ds method predicts results nearly identical to those from q1d in these steady flow cases except that it can hardly give reasonable solutions in power law fluid case with viscous exponential factor away from unity correct prediction of the stress and the wall temperature gradient is considered responsible for this better performance in simulations of the scalar convection diffusion equation it is found that the inclusion of tangential slopes in the flux computation can clearly improve the temporal accuracy of the scheme in this case the ds scheme also shows good performance the present study suggests caution in adopting of the q1d extension or ds method in multidimensional flow when it is sensitive to the accuracy of stress or wall variable gradient calculations it is better to use the md scheme for high performance simulation

further developments of the neyman scott clustered point process for modeling rainfall this paper provides some useful results for modeling rainfall it extends work on the neyman scott cluster model for simulating rainfall time series several important properties have previously been found for the model for example the expectation and variance of the amount of rain captured in an arbitrary time interval rodriguez iturbe et al 1987a in this paper additional properties are derived such as the probability of an arbitrary interval of any chosen length being dry in applications this is a desirable property to have and is often used for fitting stochastic rainfall models to field data the model is currently being used in rainfall time series research directed toward improving sewage systems in the united kingdom to illustrate the model s performance an example is given where the model is fitted to 10 years of hourly data taken from blackpool england

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

gas kinetic numerical studies of three dimensional complex flows on spacecraft re entry the gas kinetic numerical algorithm solving the boltzmann model equation is extended and developed to study the three dimensional hypersonic flows of spacecraft re entry into the atmosphere in perfect gas in this study the simplified velocity distribution function equation for various flow regimes is presented on the basis of the kinetic boltzmann shakhov model the discrete velocity ordinate technique and numerical quadrature methods such as the gauss quadrature formulas with the weight function 2 p 1 2exp v 2 and the gauss legendre numerical quadrature rule are studied to resolve the barrier in simulating complex flows from low mach numbers to hypersonic problems specially the gas kinetic finite difference scheme is constructed for the computation of three dimensional flow problems which directly captures the time evolution of the molecular velocity distribution function the gas kinetic boundary conditions and numerical procedures are studied and implemented by directly acting on the velocity distribution function the hpf high performance fortran parallel implementation technique for the gas kinetic numerical method is developed and applied to study the hypersonic flows around three dimensional complex bodies the main purpose of the current research is to provide a way to extend the gas kinetic numerical algorithm to the flow computation of three dimensional complex hypersonic problems with high mach numbers to verify the current method and simulate gas transport phenomena covering various flow regimes the three dimensional hypersonic flows around sphere and spacecraft shape with different knudsen numbers and mach numbers are studied by hpf parallel computing excellent results have been obtained for all examples computed

a three dimensional multidimensional gas kinetic scheme for the navier stokes equations under gravitational fields this paper extends the gas kinetic scheme for one dimensional inviscid shallow water equations k xu a well balanced gas kinetic scheme for the shallow water equations with source terms j comput phys 178 2002 533 562 to multidimensional gas dynamic equations under gravitational fields four important issues in the construction of a well balanced scheme for gas dynamic equations are addressed first the inclusion of the gravitational source term into the flux function is necessary second to achieve second order accuracy of a well balanced scheme the chapman enskog expansion of the boltzmann equation with the inclusion of the external force term is used third to avoid artificial heating in an isolated system under a gravitational field the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface fourth the multidimensional approach with the inclusion of tangential gradients in two dimensional and three dimensional cases becomes important in order to maintain the accuracy of the scheme many numerical examples are used to validate the above issues which include the comparison between the solutions from the current scheme and the strang splitting method the methodology developed in this paper can also be applied to other systems such as semi conductor device simulations under electric fields

application of gas kinetic scheme with kinetic boundary conditions in hypersonic flow this work focuses on the application of the gas kinetic scheme based on the bhatnagar gross krook particle collision model in the hypersonic flow simulations kinetic boundary conditions with different accommodation coefficients are constructed and implemented in the scheme the numerical study is based on a laminar shock wave boundary layer interaction in a mach 10 flow passing through a hollow cylinder flare model the grid independent numerical results agree well with experimental measurements the utilization of the kinetic slip boundary condition is necessary to improve the agreement between the current results experimental measurements as well as direct simulation monte carlo solutions the effect of the variation of the accommodation coefficient on the flow solutions is quantitatively evaluated

on the construction of kinetic schemes a general method for the construction of kinetic schemes of evolutionary equations is illustrated with the simple example of the linear advection equation where the role of the collision effect is clarified theoretically and numerically the application to the compressible euler equation and navier stokes equation is explained the theoretical backgrounds of the existing kinetic schemes such as the pullin scheme for the euler equation and chou baganoff scheme and modified prendergast xu scheme for the navier stokes equation are unified and their intrinsic errors are revealed

probabilistic representation of the temporal rainfall process by a modified neyman scott rectangular pulses model parameter estimation and validation the capability of the neyman scott clustered stochastic point process model of rainfall to preserve various observed statistics is considered randomization of the cell duration parameter from storm to storm is shown to considerably improve the wet dry period joint distribution and extreme value statistics a simple procedure for parameter estimation is introduced and applied

anomalous flow profile due to the curvature effect on slip length einzel panzer and liu phys rev lett 64 2269 1990 suggest that the slip boundary condition for a fluid moving near a wall is modified by the radius of curvature of the surface using particle simulations of a microscopic flow between concentric cylinders we qualitatively confirm their prediction and point out that the effect is seen in a limiting case derived by maxwell nature london 16 244 1877

a compressible navier stokes flow solver with scalar transport this paper concerns the extension of the gas kinetic scheme for the compressible navier stokes equations to the flow calculation with interfaces and mixing the objective of the current research targets mainly on the accurate capturing of navier stokes diffusive interfaces where the thickness can be resolved by the cell size firstly a new bgk ns scheme coupling with the level set type scalar function transport is constructed even though the scalar function is directly incorporated into the gas distribution function and it evolves according to the gas kinetic equation it likes more or less a color function which has no direct contribution for the time evolution of conservative flow variables such as mass momentum and energy due to the coupling of the scalar function into the gas kinetic formulation the governing equations for the scalar function turns out to be an advection diffusion equations and the diffusive coefficient can be controlled by the particle collision time which makes the current scheme suitable for the gas mixing problems with a controllable diffusion coefficients however for the non mixing or sharp interface problems such as the interface between gas and liquid the current method can be used as a scheme similar to the level set method where the interface location can be identified with a fixed level set value such as q 0 the current method is applied to a few examples from the simple square wave propagation and diffusion to the 3d rayleigh taylor instability the supersonic mixing layer and the shock helium bubble interaction case show clearly the convergence of the current navier stokes solver for the flow problems with mixing of components and interface once the interface thickness can be well resolved by the cell size in the case of shock hitting sf6 cylinder the computation predicts the experimental measure very well in the current scheme the schmidt number can be freely chosen according to the physical reality

statistical inference for point process models of rainfall in this paper we develop maximum likelihood procedures for parameter estimation and model selection that apply to a large class of point process models that have been used to model rainfall occurrences including cox processes neyman scott processes and renewal processes the statistical inference procedures are based on the stochastic intensity t lims 0 s 0 1 s e n t s n t n u u t the likelihood function of a point process is shown to have a simple expression in terms of the stochastic intensity the main result of this paper is a recursive procedure for computing stochastic intensities the procedure is applicable to a broad class of point process models including renewal cox process with markovian intensity processes and an important class of neyman scott processes the model selection procedure we propose which is based on likelihood ratios allows direct comparison of two classes of point processes to determine which provides a better model for a given data set the estimation and model selection procedures are applied to two data sets of simulated cox process arrivals and a data set of daily rainfall occurrences in the potomac river basin