gravity interpretation using the mellin transform the mellin transform of the gravity effect of a buried sphere and two dimensional horizontal circular cylinder and the first horizontal derivative of the gravity effect of a two dimensional thin fault layer are derived the transformed functions are bounded by two asymptotes they are analyzed and procedures are formulated excluding the asymptotic regions for the extraction of the body parameters the application of the mellin transform is tested on simulated models as well as on two field examples 1 the humble dome gravity anomaly near houston usa and 2 the louga gravity anomaly usa

stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

evaluation of the external force term in the discrete boltzmann equation a representation of the forcing term in the boltzmann equation based on a hermite expansion of the boltzmann distribution function in velocity phase space is derived based on this description of the forcing term a systematic comparison of previous lattice boltzmann models describing a nonideal gas behavior is given

discrete effects on boundary conditions for the lattice boltzmann equation in simulating microscale gas flows the lattice boltzmann equation lbe has shown its promise in the simulation of microscale gas flows one of the critical issues with this advanced method is to specify suitable slip boundary conditions to ensure simulation accuracy in this paper we study two widely used kinetic boundary conditions in the lbe the combination of the bounce back and specular reflection scheme and the discrete maxwell s scheme we show that i both schemes are virtually equivalent in principle and ii there exist discrete effects in both schemes a strategy is then proposed to adjust the parameters in the two kinetic boundary conditions such that an accurate slip boundary condition can be implemented the numerical results demonstrate that the corrected boundary conditions are robust and reliable

entropic lattice boltzmann method for microflows abstract a new method for the computation of flows at the micrometer scale is presented it is based on the recently introduced minimal entropic kinetic models both the thermal and isothermal families of minimal models are presented and the simplest isothermal entropic lattice bhatnagar gross krook elbgk is studied in detail in order to quantify its relevance for microflow simulations elbgk is equipped with boundary conditions which are derived from molecular models diffusive wall a map of three dimensional kinetic equations onto two dimensional models is established which enables two dimensional simulations of quasi two dimensional flows the elbgk model is studied extensively in the simulation of the two dimensional poiseuille channel flow results are compared to known analytical and numerical studies of this flow in the setting of the bhatnagar gross krook model the elbgk is in quantitative agreement with analytical results in the domain of weak rarefaction characterized by knudsen number kn the ratio of mean free path to the hydrodynamic scale up to kn 0 01 which is the domain of many practical microflows moreover the results qualitatively agree throughout the entire knudsen number range demonstrating knudsen s minimum for the mass flow rate at moderate values of kn as well as the logarithmic scaling at large kn the present results indicate that elbm can complement or even replace computationally expensive microscopic simulation techniques such as kinetic monte carlo and or molecular dynamics for low mach and low knudsen number hydrodynamics pertinent to microflows

analysis of depth resolution in potential field inversion we study the inversion of potential fields and evaluate the degree of depth resolution achievable for a given problem to this end we introduce a powerful new tool the depth resolution plot drp the drp allows a theoretical study of how much the depth resolution in a potential field inversion is influenced by the way the problem is discretized and regularized the drp also allows a careful study of the influence of various kinds of ambiguities such as those from data errors or of a purely algebraic nature the achievable depth resolution is related to the given discretization regularization and data noise level we compute drp by means of singular value decomposition svd or its generalization gsvd depending on the particular regularization method chosen to illustrate the use of the drp we assume a source volume of specified depth and horizontal extent in which the solution is piecewise constant within a 3d grid of blocks we consider various linear regularization terms in a tikhonov damped least squares formulation some based on using higher order derivatives in the objective function drps are illustrated for both synthetic and real data our analysis shows that if the algebraic ambiguity is not too large and a suitable smoothing norm is used some depth resolution can be obtained without resorting to any subjective choice of depth weighting

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

fitzpatrick functions and continuous linear monotone operators the notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex lower semicontinuous and proper function and any not necessarily symmetric continuous linear positive operator it was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing fitzpatrick functions in this paper we study fitzpatrick functions of continuous linear monotone operators defined on a hilbert space a novel characterization of skew operators is presented a result by brezis and haraux is reproved using the fitzpatrick function we investigate the fitzpatrick function of the sum of two operators and we show that a known upper bound is actually exact in finite dimensional and more general settings cyclic monotonicity properties are also analyzed and closed forms of the fitzpatrick functions of all orders are provided for all rotators in the euclidean plane

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa

lattices for the lattice boltzmann method a recently introduced theory of higher order lattice boltzmann models chikatamarla and karlin phys rev lett 97 190601 2006 is elaborated in detail a general theory of the construction of lattice boltzmann models as an approximation to the boltzmann equation is presented new lattices are found in all three dimensions and are classified according to their accuracy degree of approximation of the boltzmann equation the numerical stability of these lattices is argued based on the entropy principle the efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions

the resolvent average of monotone operators dominant and recessive properties within convex analysis a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago when one considers the subdifferential of the proximal average a natural averaging operation of the subdifferentials of the averaged functions emerges in the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in hilbert spaces transforming it into the resolvent average the theory of resolvent averages contains many desirable properties in particular we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average as a consequence we not only recover a significant part of the theory of proximal averages but also obtain results and properties which have not been previously available

implementation of diffuse reflection boundary conditions in a thermal lattice boltzmann model with flux limiters we discuss three new implementation versions of diffuse reflection boundary conditions in a thermal lattice boltzmann model their accuracy is investigated in the case of couette flow by considering the slip regime the best results are recovered with versions 2 and 3 which rely on outgoing fluxes to express the particle distribution functions in the ghost nodes outside the flow domain version 2 is found to be more economical since it involves no interpolation procedure this version was thereafter used to investigate the temperature profile in couette flow for various values of prandtl number as well as the capability of the thermal lb model to capture the knudsen minimum in poiseuille flow

accuracy of higher order lattice boltzmann methods for microscale flows with finite knudsen numbers accuracy of the lattice boltzmann lb method for microscale flows with finite knudsen numbers is investigated we employ up to the eleventh order gauss hermite quadrature for the lattice velocities and diffuse scattering boundary condition for fluid wall interactions detailed comparisons with the direct simulation monte carlo dsmc method and the linearized boltzmann equation are made for planar couette and poiseuille flows all higher order lb methods considered here give improved results as compared with the standard lb method the accuracy of the lb hierarchy however does not monotonically increase with the order of the gauss hermite quadrature at moderate and large knudsen numbers the results also show the sensitivity to a quadrature chosen even when the gauss hermite quadratures have the same order of formal accuracy among the schemes investigated here d2q16 is the most efficient method and offers a quantitative prediction in the slip and transition regimes the higher order lb methods predict the knudsen layer up to kn o 0 1 the knudsen layer however rapidly disappears when the knudsen number approaches unity due to a finite number of the lattice velocities while it is still present for kn o 1 in the boltzmann equation it is also found that the higher order lb methods adopted here do not capture the asymptotic behavior of the boltzmann equation at large knudsen numbers

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

renormalization of the lattice boltzmann hierarchy is it possible to solve boltzmann type kinetic equations using only a small number of particle velocities we introduce a technique of solving kinetic equations with a arbitrarily large number of particle velocities using only a lattice boltzmann method on standard low symmetry lattices the renormalized kinetic equation is validated with an exact solution of the planar couette flow at moderate knudsen numbers

asplund decomposition of monotone operators we establish representations of a monotone mapping as the sum of a maximal subdifferential mapping and a remainder monotone mapping where the remainder is acyclic in the sense that it contains no nontrivial subdifferential component this is the nonlinear analogue of a skew linear operator examples of indecomposable and acyclic operators are given in particular we present an explicit nonlinear acyclic operator

exploring the klinkenberg effect at different scales simulations of microflows usually require sophisticated numerical tools nevertheless in the slip regime the hydrodynamic equation with slip boundary condition may be sufficient to account for the so called klinkenberg effect we propose to visit this effect using a basic network of microchannels in which the knudsen number is multiplied by two or four by introducing successive derivations to the channel we derived an equivalent hydraulic conductivity up to second order theoretical results are compared both with the results of the navier stokes equations with slip condition and with those obtained using a bhatnagar gross krook hermite model developed especially for flows with a large spectrum of knudsen numbers typically 10 4 k n 10 a criterion is provided in order to distinguish the slip regime from the transitional one in this multiscale network

on the uniqueness of the inverse potential problem for homogeneous polyhedrons the uniqueness of a homogeneous polyhedron as determined from its exterior newtonian potential has been studied as a result of the special apparatus developed in this paper the two dimensional research methods could be applied to study three dimensional cases

high order lattice boltzmann models for wall bounded flows at finite knudsen numbers we analyze a large number of high order discrete velocity models for solving the boltzmann bhatnagar gross krook equation for finite knudsen number flows using the chapman enskog formalism we prove for isothermal flows a relation identifying the resolved flow regimes for low mach numbers although high order lattice boltzmann models recover flow regimes beyond the navier stokes level we observe for several models significant deviations from reference results we found this to be caused by their inability to recover the maxwell boundary condition exactly by using supplementary conditions for the gas surface interaction it is shown how to systematically generate discrete velocity models of any order with the inherent ability to fulfill the diffuse maxwell boundary condition accurately both high order quadratures and an exact representation of the boundary condition turn out to be crucial for achieving reliable results for poiseuille flow we can reproduce the mass flow and slip velocity up to the knudsen number of 1 moreover for small knudsen numbers the knudsen layer behavior is recovered

consistent second order boundary implementations for convection diffusion lattice boltzmann method in this study an alternative second order boundary scheme is proposed under the framework of the convection diffusion lattice boltzmann lb method for both straight and curved geometries with the proposed scheme boundary implementations are developed for the dirichlet neumann and linear robin conditions in a consistent way the chapman enskog analysis and the hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second order accuracy then the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming see the paragraph after eq 29 for the discussions of the streaming step in lb method after that the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes for straight boundaries boundary nodes are directly placed at the physical boundary surface and the present scheme is applied directly when extending the present scheme to curved geometries a local curvilinear coordinate system and first order taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface in essence the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface therefore the advantages of the present boundary implementations are i the locality i e no information from neighboring fluid nodes is required ii the consistency i e the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes thus the three kinds of conditions are realized in a consistent way it should be noted that the present focus is on two dimensional cases and theoretical derivations as well as the numerical validations are performed in the framework of the two dimensional five velocity lattice model

an explicit example of a maximal 3 cyclically monotone operator with bizarre properties abstract subdifferential operators of proper convex lower semicontinuous functions and more generally maximal monotone operators are ubiquitous in optimization and nonsmooth analysis in between these two classes of operators are the maximal n cyclically monotone operators these operators were carefully studied by asplund who obtained a complete characterization within the class of positive semidefinite not necessarily symmetric matrices and by voisei who presented extension theorems a la minty all previous explicit examples of maximal n cyclically monotone operators are maximal monotone thus they inherit the known good properties of maximal monotone operators in this paper we construct an explicit maximal 3 cyclically monotone operator with quite bizarre properties this construction builds upon a recent nonconstructive and zorn s lemma based example our operator possesses two striking properties that sets it far apart from both the maximal monotone operator and the subdifferential operator case it is not maximal monotone and its domain which is closed fails to be convex indeed the domain is the boundary of the unit diamond in the euclidean plane the path leading to this operator requires some new results that are interesting in their own right

hydrodynamics beyond navier stokes the slip flow model recently analytical solutions for the nonlinear couette flow demonstrated the relevance of the lattice boltzmann lb models to hydrodynamics beyond the continuum limit s ansumali et al phys rev lett 98 124502 2007 in this paper we present a systematic study of the simplest lb kinetic equation the nine bit model in two dimensions in order to quantify it as a slip flow approximation details of the aforementioned analytical solution are presented and results are extended to include a general shear and force driven unidirectional flow in confined geometry exact solutions for the velocity as well as for pertinent higher order moments of the distribution functions are obtained in both couette and poiseuille steady state flows for all values of rarefaction parameter knudsen number results are compared with the slip flow solution by cercignani and a good quantitative agreement is found for both flow situations thus the standard nine bit lb model is characterized as a valid and self consistent slip flow model for simulations beyond the navier stokes approximation

lattice boltzmann models based on half range gauss hermite quadratures we discuss general features of thermal lattice boltzmann models based on half range gauss quadratures specialising to the half range gauss hermite and gauss laguerre cases the main focus of the paper is on the construction of high order half range hermite lattice boltzmann hhlb models the performance of the hhlb models is compared with that of laguerre lattice boltzmann llb and full range hermite lattice boltzmann hlb models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density macroscopic velocity temperature and heat fluxes in the two dimensional couette flow the bhatnagar gross krook bgk collision term is used throughout the paper to reduce the computational costs of the numerical simulations we use mixed lattice boltzmann models constructed using different quadrature methods on each cartesian axis for kn 0 01 the hlb models require the least number of velocities to satisfy our convergence test when kn 0 05 the hlb models are outperformed in terms of number of velocities employed by both the llb and the hhlb models moreover we find that the hhlb models require less quadrature points than the llb models at all tested values of kn which we attribute to the maxwellian form of the weight function for the half range hermite polynomials

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

coupled double distribution function lattice boltzmann method for the compressible navier stokes equations a coupled double distribution function lattice boltzmann method is developed for the compressible navier stokes equations different from existing thermal lattice boltzmann methods this method can recover the compressible navier stokes equations with a flexible specific heat ratio and prandtl number in the method a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations while the compressible energy equation is recovered by an energy distribution function the energy distribution function is then coupled to the density distribution function via the thermal equation of state in order to obtain an adjustable specific heat ratio a constant related to the specific heat ratio is introduced into the equilibrium energy distribution function two different coupled double distribution function lattice boltzmann models are also proposed in the paper numerical simulations are performed for the riemann problem the double mach reflection problem and the couette flow with a range of specific heat ratios and prandtl numbers the numerical results are found to be in excellent agreement with analytical and or other solutions

boundary conditions for the upwind finite difference lattice boltzmann model evidence of slip velocity in micro channel flow we conduct a systematic study of the effect of various boundary conditions bounce back and three versions of diffuse reflection for the two dimensional first order upwind finite difference lattice boltzmann model simulation of couette flow in a micro channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the knudsen number e l l where l is the mean free path and l is the channel width for walls moving in opposite directions with speeds u w the slip velocity satisfies u s l i p 2 eu w a l l 1 2 e in the case of poiseuille flow in a micro channel the slip velocity is found to depend on the lattice spacing ds and knudsen number e to both first and second order the best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes

simulation of micro and nano scale flows via the lattice boltzmann method abstract we use the lattice boltzmann method lbm for analysis of high and moderate knudsen number phenomena simulation results are presented for microscale couette and poiseuille flows the slip velocity nonlinear pressure drop and mass flow rate are compared with previous numerical results and or experimental data the knudsen minimum is successfully predicted for the first time within the lbm framework these results validate the usage of the lbm based commercial arbitrary geometry code powerflow for simulating nanoscale problems

gravity inversion using open reject and shape of anomaly fill criteria a gravity inversion method is developed by iteratively applying open reject and fill o r f criteria within a model space comprising a great many rectangular prisms each prism is assigned a density contrast the modeling procedure consists of filling some prisms while leaving others empty only one element is filled for each pass generally elements are added only to the periphery of the growing model models can be allowed to grow in any combination of directions or in all directions by application of a shape of anomaly fill criterion the model rapidly attains a form which yields gravity approximating the given gravity scaled down by some constant factor as the model continues to grow this scale factor approaches unity the method readily yields inverse models comprising several thousand individual prisms examples presented here give applications to 2 d problems the method is readily applicable to 212 d and 3 d problems as well overhanging elements are obtained by appropriate use of model

autoconjugate representers for linear monotone operators monotone operators are of central importance in modern optimization and nonlinear analysis their study has been revolutionized lately due to the systematic use of the fitzpatrick function pioneered by penot and svaiter a topic of recent interest has been the representation of maximal monotone operators by so called autoconjugate functions two explicit constructions were proposed the first by penot and z linescu in 2005 and another by bauschke and wang in 2007 the former requires a mild constraint qualification while the latter is based on the proximal average we show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces the continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate furthermore we also construct an infinite family of autoconjugate representers for the identity operator on the real line

fundamental conditions for n th order accurate lattice boltzmann models in this paper we theoretically prove a set of fundamental conditions pertaining to discrete velocity sets and corresponding weights these conditions provide sufficient conditions for a priori formulation of lattice boltzmann models that automatically admit correct hydrodynamic moments up to any given n th order

two dimensional thermal model of the finite difference lattice boltzmann method with high spatial isotropy the existing lattice boltzmann method multispeed thermal models show a limited accuracy this paper proposes a two dimensional multispeed thermal model for the finite difference lattice boltzmann method fdlbm to recover correct fluid equations up to fourth orders of local flow velocity should be retained in the local equilibrium distribution function and tensors of particle velocities should have up to seventh rank isotropy in the fdlbm particle velocities can be selected independently from the lattice configuration therefore particle velocities of octagonal directions which have up to seventh rank isotropic tensors are adopted the proposed model was verified by two simulations the model showed excellent numerical stability in addition to strict accuracy

an alternative implementation of the kinetic theory based axisymmetric lattice boltzmann model abstract the semi implicit centered scheme as an approximated implementation of the trapezium rule based scheme has proved to be capable of avoiding the discrete lattice effects in our recent forcing scheme analysis for the axisymmetric lb method and is applied here for an alternative implementation of the kinetic theory based axisymmetric model by guo et al in particular the external force terms and the first order part of the source terms accounting for the axisymmetric effect are discretized with the semi implicit centered scheme while the rest are represented with the trapezium rule the proposed axisymmetric lb scheme has the advantage of simpler calculations of the macroscopic variables since the zeroth and the first order moments of the distribution function are independent of the complicated source terms chapman enskog analysis in the appendix demonstrates that axisymmetric navier stokes n s equations are recovered from the proposed model and the error terms caused by the semi implicit centered scheme do not affect the overall accuracy of the present model moreover the accuracy and applicability of the proposed axisymmetric lb model are verified by the adopted numerical tests and the complicated macroscopic variables computations in the guo et al model and its derivative models are avoided by the present model

some progress in lattice boltzmann method part i nonuniform mesh grids a new lattice boltzmann algorithm is proposed to simulate the navier stokes equation on arbitrary nonuniform mesh grids the new algorithm retains the advantages of the lattice boltzmann method parallel of algorithm ease of programming and ability to incorporate microscopic interactions a simulation of flow in a two dimensional symmetric channel with sudden expansion is carried out using the new algorithm on a nonuniform mesh the results of the simulation are in excellent agreement with previous experimental and numerical results

convergence of a discrete velocity model for the boltzmann bgk equation abstract we prove the convergence of a conservative and entropic discrete velocity model for the bathnagar gross krook bgk equation in this model the approximation of the maxwellian is based on a discrete entropy minimization principle the main difficulty due to its implicit definition is to prove that this approximation is consistent we also demonstrate the existence and uniqueness of a solution to the discrete velocity model by using a fixed point theorem finally the model is written in a continuous equation form and we prove the convergence of its solution toward a solution of the bgk equation

practical applications of uniqueness theorems in gravimetry part i constructing sound interpretation methods to obtain a unique and stable solution to the gravity inverse problem a priori information reflecting geological attributes of the gravity source must be used mathematical conditions to obtain stable solutions are established in tikhonov s regularization method where the a priori information is introduced via a stabilizing functional which may be suitably designed to incorporate some relevant geological information however there is no unifying approach establishing general uniqueness conditions for a gravity inverse problem rather there are many theorems usually establishing just abstract mathematical conditions and making it difficult to devise the type of geological information needed to guarantee a unique solution in part i of these companion papers we show that translating the mathematical uniqueness conditions into geological constraints is an important step not only in establishing the type of geological setting where a particular method may be applied but also in designing new gravity inv

the lattice boltzmann equation on irregular lattices a general framework to extend the lattice boltzmann equation to arbitrary lattice geometries is presented and numerically demonstrated for the case of a two dimensional poiseuille flow the new scheme considerably extends the range of applicability of the boltzmann method to problems requiring the use of nonuniform grids

practical applications of uniqueness theorems in gravimetry part ii pragmatic incorporation of concrete geologic information we illustrate the importance of establishing solution uniqueness through mathematical restrictions reflecting a source attribute we also illustrate the validity and utility of a guideline derived in an accompanying paper for constructing sound gravity inversion methods for the class of sources presenting either homogeneous or depth independent density distributions the two part guideline is 1 to introduce a priori information favoring uniqueness either by assuming that a nonnull density distribution depending only on x and y is confined to the interior of a horizontal slab with known position or by limiting the class of possible solutions to homogeneous simply connected polygons or polyhedra with known density displaying no fancy shapes and no curling apophyses at their borders and 2 to introduce information favoring solution stability by estimating only the features of the source which may be resolved by the data following the guideline we apply different methods to gravity data using interp

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

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

poiseuille type flow of a rarefied gas between two parallel plates driven by a uniform external force a unidirectional flow of a rarefied gas between two parallel plates driven by a uniform external force is investigated on the basis of kinetic theory with special interest in the behavior in the near continuum regime the bhatnagar gross krook bgk model of the boltzmann equation and the diffuse reflection boundary condition are employed as the basic system first a systematic asymptotic analysis of the basic system for small knudsen numbers is carried out and a system of fluid dynamic type equations and their boundary conditions are derived up to the second order in the knudsen number then an accurate numerical analysis of the original bgk system is performed for a wide range of the knudsen number by means of a finite difference method the behavior of the gas such as the non navier stokes effects in the near continuum regime is clarified on the basis of the fluid dynamic type system as well as the numerical solution of the bgk system

nearly convex sets fine properties and domains or ranges of subdifferentials of convex functions nearly convex sets play important roles in convex analysis optimization and theory of monotone operators we give a systematic study of nearly convex sets and construct examples of subdifferentials of lower semicontinuous convex functions whose domains or ranges are nonconvex

diffuse reflection boundary condition for high order lattice boltzmann models with streaming collision mechanism for lattice boltzmann lb models their particle feature is ensured by the streaming collision mechanism however this mechanism is often discarded for high order lb models due to multi speed lattices here we propose a new way of implementing the diffuse reflection boundary condition that can maintain the characteristic streaming collision mechanism for high order models it is then tested on four lattice models namely d2q16 d2q17 d2q37 and d3q121 for isothermal and thermal couette flows and lid driven cavity flows our implementation is shown to be able to achieve second order accuracy globally

kinetic boundary conditions in the lattice boltzmann method derivation of the lattice boltzmann method from the continuous kinetic theory x he and l s luo phys rev e 55 r6333 1997 x shan and x he phys rev lett 80 65 1998 is extended in order to obtain boundary conditions for the method for the model of a diffusively reflecting moving solid wall the boundary condition for the discrete set of velocities is derived and the error of the discretization is estimated numerical results are presented which demonstrate convergence to the hydrodynamic limit in particular the knudsen layer in the kramers problem is reproduced correctly for small knudsen numbers

analytic solution for a higher order lattice boltzmann method slip velocity and knudsen layer we present the analysis of a higher order lattice boltzmann lb method based on the fourth order gauss hermite quadrature with emphasis on the slip velocity and the knudsen layer the exact solution of the slip velocity for the higher order lb equation is obtained for poiseuille flows with finite knudsen numbers due to increased accuracy in velocity space discretization the higher order scheme gives much improved slip coefficients as compared with the standard lb method based on the third order gauss hermite quadrature a multiple relaxation time model is investigated to show the effects of the relaxation times for higher order moments on the slip phenomena

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

efficient kinetic method for fluid simulation beyond the navier stokes equation we present a further theoretical extension to the kinetic theory based formulation of the lattice boltzmann method of shan j fluid mech 550 413 2006 in addition to the higher order projection of the equilibrium distribution function and a sufficiently accurate gauss hermite quadrature in the original formulation a regularization procedure is introduced in this paper this procedure ensures a consistent order of accuracy control over the nonequilibrium contributions in the galerkin sense using this formulation we construct a specific lattice boltzmann model that accurately incorporates up to third order hydrodynamic moments numerical evidence demonstrates that the extended model overcomes some major defects existing in conventionally known lattice boltzmann models so that fluid flows at finite knudsen number kn can be more quantitatively simulated results from force driven poiseuille flow simulations predict the knudsen s minimum and the asymptotic behavior of flow flux at large kn

analytical solution of axi symmetrical lattice boltzmann model for cylindrical couette flows abstract analytical solution for the axi symmetrical lattice boltzmann model is obtained for the low mach number cylindrical couette flows in the hydrodynamic limit the present solution is in excellent agreement with the result of the navier stokes equation since the kinetic boundary condition is used the present analytical solution using nine discrete velocities can describe flows with the knudsen number up to 0 1 meanwhile the comparison with the simulation data obtained by the direct simulation monte carlo method shows that higher order lattice boltzmann models with more discrete velocities are needed for highly rarefied flows

examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by b f svaiter abstract in this paper we give two explicit examples of unbounded linear maximal monotone operators the first unbounded linear maximal monotone operator s on l 2 is skew we show its domain is a proper subset of the domain of its adjoint s and s is not maximal monotone this gives a negative answer to a recent question posed by svaiter the second unbounded linear maximal monotone operator is the inverse volterra operator t on l 2 0 1 we compare the domain of t with the domain of its adjoint t and show that the skew part of t admits two distinct linear maximal monotone skew extensions these unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened and they are simpler than the example given by phelps simons interesting consequences on fitzpatrick functions for sums of two maximal monotone operators are also given

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

the brezis browder theorem in a general banach space during the 1970s brezis and browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces in this paper we extend and refine their result to a general banach space we also provide an affirmative answer to a problem posed by phelps and simons

boundary conditions for thermal lattice boltzmann equation method we propose a thermal boundary condition treatment based on the bounce back idea and interpolation of the distribution functions for both the dirichlet and neumann normal derivative conditions in the thermal lattice boltzmann equation tlbe method the coefficients for the distribution functions involved are determined to satisfy the dirichlet or neumann condition with second order accuracy for the dirichlet condition there is an adjustable parameter in the treatment and three particular schemes are selected for demonstration while for the neumann condition the second order accurate scheme is unique when applied to inclined or curved boundaries the dirichlet condition treatment can be directly used while the neumann condition given in the normal direction of the boundary should be converted into derivative conditions in the discrete velocity directions of the tlbe model a spatially coupled formula relating the boundary temperature boundary normal heat flux and the distribution functions near the boundary is thus derived for the neumann problems on curved boundaries the applicability and accuracy of the present boundary treatments are examined with several numerical tests for which analytical solutions are available including the 2 dimensional 2 d steady state channel flow the 1 d transient heat conduction in an inclined semi infinite solid the 2 d steady state and transient heat conduction inside a circle and the 3 d steady state circular pipe flow while the dirichlet condition treatment leads to second order accuracy for the temperature field it only gives a first order accurate boundary heat flux because of the irregularity of the cuts by the curved boundary with the lattices with the neumann condition on the curved boundary the accuracy for the temperature field obtained is first order when the tangential temperature gradient on the boundary is decoupled second order convergence of the temperature field can be obtained with neumann conditions

the kernel average for two convex functions and its application to the extension and representation of monotone operators we provide and analyze an average for two convex functions based on a kernel function it covers several known averages such as the arithmetic average epigraphical average and the proximal average when applied to the fitzpatrick function and the conjugate of fitzpatrick function associated with a monotone operator our average produces an autoconjugate also known as selfdual lagrangian which can be used for finding an explicit maximal monotone extension of the given monotone operator this completely settles one of the open problems posed by fitzpatrick in the setting of reflexive banach spaces

numerical simulation of flows from free molecular regime to continuum regime by a dvm with streaming and collision processes a discrete velocity method dvm with streaming and collision processes is presented in this work for simulation of flows from free molecular regime to continuum regime the present scheme can be considered as a semi lagrangian like scheme at first we follow the conventional dvm to discretize the phase velocity space by a number of discrete velocities then for each discrete velocity the kinetic equation with bgk shakhov model is integrated in space and time within one time step as a result a simple algebraic formulation can be obtained and its solution can be marched in time by the streaming and collision processes however differently from the conventional semi lagrangian scheme the present scheme uses the muscl approach with van albada limiter in the process of reconstructing the distribution function at the surrounding points of the cell center and the transport distance is controlled in order to avoid extrapolation this makes the present scheme be capable of simulating the hypersonic rarefied flows in addition as compared to the unified gas kinetic scheme ugks the present scheme is simpler and easier for implementation thus the computational efficiency can be improved accordingly to validate the proposed numerical scheme test examples from free molecular regime to continuum regime are simulated numerical results showed that the present scheme can predict the flow properties accurately even for hypersonic rarefied flows

finite difference lattice boltzmann method with arbitrary specific heat ratio applicable to supersonic flow simulations abstract the current finite difference lattice boltzmann method fdlbm gives a fixed specific heat ratio because internal energy is limited to the translational freedom of the space yan et al and kataoka et al clarified the conditions for deriving models with arbitrary specific heat ratio and proposed euler models however these model applications to numerical simulations showed the weakness in the numerical stability in this paper a two dimensional fdlbm navier stokes model and a three dimensional fdlbm euler model which allow arbitrary values to be set for the specific heat ratio were proposed these models stably performed numerical simulations from subsonic to supersonic ranges

twosetsofnewcharacterizationsfornormaland ep matrices two sets of new characterizations for normal matrices and ep matrices are presented which are derived through ranks and generalized inverses of matrices

stabilization of the lattice boltzmann method by the h theorem a numerical test for a one dimensional benchmark shock tube problem we implement the lattice boltzmann method based on the h theorem i karlin a ferrante and h c ottinger europhys lett 47 182 1999 results of simulation demonstrate significant improvement of stability as compared to realizations without explicit entropic estimations

slip velocity and knudsen layer in the lattice boltzmann method for microscale flows we present mesoscopic fluid wall interaction models for lattice boltzmann lb model simulations of microscale flows the exact solution of the slip velocity for the lb equation with the bhatnagar gross krook collision operator is obtained for poiseuille flow at finite knudsen numbers with a consistent definition of the knudsen number the slip coefficients of the lb equation with the standard d2q9 scheme are found to be slightly larger than those of the boltzmann equation with the same boundary condition which makes the standard lb method remain quantitatively accurate only for small knudsen numbers by modifying the nonequilibrium energy flux or introducing the effective relaxation time the lb method is analytically shown to reproduce the slip phenomena up to second order in the knudsen number for the standard lb method the knudsen layer is captured only with modification of the relaxation dynamics such as in the effective relaxation time model

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

primal dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators a fundamental result due to rockafellar states that every cyclically monotone operator a admits an antiderivative f in the sense that the graph of a is contained in the graph of the subdifferential operator partial f given a method mathfrak m that assigns every finite cyclically monotone operator a some antiderivative mathfrak m a we say that the method is primal dual symmetric if mathfrak m applied to the inverse of a produces the fenchel conjugate of mathfrak m a rockafellar s antiderivatives do not possess this property utilizing fitzpatrick functions and the proximal average we present novel primal dual symmetric intrinsic methods the antiderivatives produced by these methods provide a solution to a problem posed by rockafellar in 2005 the results leading to this solution are illustrated by various examples

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level