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

comparisons of lattice boltzmann and finite difference methods for a two dimensional viscous burgers equation lattice boltzmann methods have been proposed as a computational means for solving various partial differential equations we look at three lattice boltzmann methods and an analogous finite difference method for solving a two dimensional scalar viscous burgers equation with periodic boundary conditions first using monotonicity arguments we prove that in the ell 1 norm the lattice boltzmann methods converge first order temporally and second order spatially then we provide some computational results which substantiate the theoretical results finally we compare the lattice boltzmann and finite difference methods the basis of comparison includes numerical results with regard to accuracy order of convergence performance and timing measurements memory requirements and conservations laws

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

semi implicit linearized multiple relaxation time formulation of lattice boltzmann schemes for mixture modeling a lattice boltzmann model for mixture modeling is developed by applying the multiple relaxation time mrt approach to the hamel model which allows one to derive from a general framework different model equations independently proposed like the gross krook model and the sirovich model by imposing some physical constraints the mrt lattice boltzmann hamel model reduces to the generalized mrt lattice boltzmann gross krook model involving the local maxwellian centered on the barycentric velocity which allows one to tune independently the species diffusivity the mixture kinematic viscosity and the mixture bulk viscosity reducing the number of moving particles over the total is possible to deal effectively with mass particle ratios far from unity and for this reason to model the pressure driven diffusion a convenient numerical approach is proposed for solving the developed model which essentially widens the stability range of conventional schemes in terms of dimensionless relaxation frequencies by solving explicitly the advection operator together with the nonlinear terms of the collisional operator and solving implicitly the residual linear terms in this way the calculations are drastically reduced and the operative matrices can be computed once for all at the beginning of the calculation implying moderate additional computational demand following this approach a semi implicit linearized backward euler scheme ideal for parallel implementations is proposed in order to achieve the previous results the asymptotic analysis recently suggested for analyzing the macroscopic equations corresponding to lattice boltzmann schemes in the low mach number limit proves to be an effective tool some numerical tests are reported for proving the consistency of the proposed method with both the fick model and maxwell stefan model in the macroscopic limit

an improved algorithm for the evaluation of cauchy principal value integrals of oscillatory functions and its application abstract a new interpolatory type quadrature rule is proposed for the numerical evaluation of cauchy principal value integrals of oscillatory kind 1 1 f x x e i x d x where 1 1 the method is based on an interpolatory procedure at clenshaw curtis points and the singular point and the fast computation of the modified moments with cauchy type singularity based on this result a new method is presented for the computation of the oscillatory integrals with logarithmic singularities too these methods enjoy fast implementation and high accuracy convergence rates on are also provided numerical examples support the theoretical analyses

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

uniform approximation to finite hilbert transform of oscillatory functions and its algorithm abstract for the finite hilbert transform of oscillatory functions q f c 1 1 f x e i x x c d t with a smooth function f and real 0 for c 1 1 in the sense of cauchy principal value or for c 1 of hadamard finite part we present an approximation method of clenshaw curtis type and its algorithm interpolating f by a polynomial p n of degree n and expanding in terms of the chebyshev polynomials with o n log n operations by the fft we obtain an approximation q p n c q f c we write q p n c as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree n 1 we efficiently evaluate the oscillatory integral with a combination of authors previous method and keller s method for f z analytic on the interval 1 1 in the complex plane z the error of q p n c is bounded uniformly with respect to c and numerical examples illustrate the performance of our method

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

constructive linear time temporal logic proof systems and kripke semantics in this paper we study a version of constructive linear time temporal logic ltl with the next temporal operator the logic is originally due to davies who has shown that the proof system of the logic corresponds to a type system for binding time analysis via the curry howard isomorphism however he did not investigate the logic itself in detail he has proved only that the logic augmented with negation and classical reasoning is equivalent to the next fragment of the standard formulation of classical linear time temporal logic we give natural deduction sequent calculus and hilbert style proof systems for constructive ltl with conjunction disjunction and falsehood and show that the sequent calculus enjoys cut elimination moreover we also consider kripke semantics and prove soundness and completeness one distinguishing feature of this logic is that distributivity of the next operator over disjunction a b a b is rejected in view of a type theoretic interpretation

fast computation of a class of highly oscillatory integrals in this paper the problem of numerical evaluation of a class of highly oscillatory integrals a b f x x a b x k 1 n x k k e i x dx where 1 k 1 a k b and 1 has been discussed based on analytic continuation the integrals are transformed into the problems of integrating the forms on 0 with the integrand does not oscillate and decays exponentially fast which can be efficiently computed by using general gauss laguerre quadrature rule theoretical results and numerical experiments perform that the method is very efficient in obtaining very high precision approximations if is sufficiently large

uniform approximation to fractional derivatives of functions of algebraic singularity fractional derivative d qf x 0 1 with g x being a well behaved function we propose a quadrature method for uniformly approximating d q x ag x the present method consists of interpolating g x at abscissae in 0 1 by a finite sum of chebyshev polynomials it is shown that the use of the lower endpoint x 0 as an abscissa is essential for the uniform approximation namely to bound the approximation errors independently of x 0 1 numerical examples demonstrate the performance of the present method

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

high order upwind compact finite difference lattice boltzmann method for viscous incompressible flows abstract in this work a high order upwind compact finite difference lattice boltzmann method ucdlbm is developed to efficiently solve viscous incompressible flow problems a fifth order upwind compact difference scheme is adopted to discretize the spatial derivatives of the lattice boltzmann equation and the third order total variation diminishing runge kutta scheme is utilized for the discretization of the temporal term compared to the existing central compact finite difference lattice boltzmann method cfdlbm the present ucdlbm can prevent non physical oscillations without filtering due to the natural dissipative property of upwind schemes three benchmark problems involving the taylor green vortex problem the doubly periodic shear layer flow problem and the lid driven square cavity flow problem are numerically solved to demonstrate the accuracy and efficiency of the present method numerical results computed are in good agreement with the analytical solution or other available numerical results and the present ucdlbm is less time consuming than the cfdlbm without degenerating the order of accuracy of the numerical solutions

propositional lax logic we investigate a peculiar intuitionistic modal logic called propositional lax logic pll which has promising applications to the formal verification of computer hardware the logic has emerged from an attempt to express correctness up to behavioural constraints a central notion in hardware verification as a logical modality as a modal logic it is special since it features a single modal operator that has a flavour both of possibility and of necessity in the paper we provide the motivation for pll and present several technical results we investigate some of its proof theoretic properties presenting a cut elimination theorem for a standard gentzen style sequent presentation of the logic we go on to define a new class of fallible two frame kripke models for pll these models are unusual since they feature worlds with inconsistent information furthermore the only frame condition imposed is that the frame be a subrelation of the frame we give a natural translation of these models into goldblatt s j space models of pll our completeness theorem for these models yields a godel style embedding of pll into a classical bimodal theory of type s4 s4 and underpins a simple proof of the finite model property we proceed to prove soundness and completeness of several theories for specialized classes of models we conclude with a brief exploration of two concrete and rather natural types of model from hardware verification for which the modality models correctness up to timing constraints we obtain decidability of free fragment of the logic of the first type of model which coincides with the stable form of maksimova s intermediate logicl

on the evaluation of cauchy principal value integrals of oscillatory functions the problem of the numerical evaluation of cauchy principal value integrals of oscillatory functions 1 1e i w xf x x tdx where 1 t 1 has been discussed based on analytic continuation if f is analytic in a sufficiently large complex region g containing 1 1 the integrals can be transformed into the problems of integrating two integrals on 0 with the integrand that does not oscillate and that decays exponentially fast which can be efficiently computed by using the gauss laguerre quadrature rule the validity of the method has been demonstrated in the provision of two numerical experiments and their results

quadrature formulae for cauchy principal value integrals of oscillatory kind etude de l evaluation numerique d une integrale 7b f 1 1 exp i x f x dx ou f possede un pole simple dans l intervalle 1 1 application de la formule d interpolation de lagrange estimation de l erreur et exemples numeriques

evaluation of cauchy principal value integrals of oscillatory kind abstract this paper studies the evaluation of cauchy principal value c p v integrals of oscillatory kind a b f x x e i x d x where a b the singularity of the c p v integral is transferred to an individual oscillatory integral that can be analytically calculated the remaining integral which is non singular can be well calculated with a stable and fast quadrature method namely the improved levin method numerical examples confirm the advantages of the proposed method

asymptotic expansions and fast computation of oscillatory hilbert transforms in this paper we study the asymptotics and fast computation of the one sided oscillatory hilbert transforms of the form begin aligned h f t e i omega t x int nolimits 0 infty e i omega t frac f t t x dt quad omega 0 quad x ge 0 end aligned where the bar indicates the cauchy principal value and f is a real valued function with analytic continuation in the first quadrant except possibly a branch point of algebraic type at the origin when x 0 the integral is interpreted as a hadamard finite part integral provided it is divergent asymptotic expansions in inverse powers of omega are derived for each fixed x ge 0 which clarify the large omega behavior of this transform we then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms depending on the position of x we classify our discussion into three regimes namely x mathcal o 1 or x gg 1 0 x ll 1 and x 0 numerical experiments show that the convergence of the proposed methods greatly improve when the frequency omega increases some extensions to oscillatory hilbert transforms with bessel oscillators are briefly discussed as well

an embedded formula of the chebyshev collocation method for stiff problems abstract in this study we have developed an embedded formula of the chebyshev collocation method for stiff problems based on the zeros of the generalized chebyshev polynomials a new strategy for the embedded formula using a pair of methods to estimate the local truncation error as performed in traditional embedded runge kutta schemes is proposed the method is performed in such a way that not only the stability region of the embedded formula can be widened but by allowing the usage of larger time step sizes the total computational costs can also be reduced in terms of concrete convergence and stability analysis the constructed algorithm turns out to have an 8th order convergence and it exhibits a stability through several numerical experimental results we have demonstrated that the proposed method is numerically more efficient compared to several existing implicit methods

analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model in this paper we analytically solve the velocity of the lattice boltzmann bgk equation lbgk for several simple flows the analysis provides a framework to theoretically analyze various boundary conditions in particular the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition we find that the slip velocity is zero as long as f e 0 at boundaries no matter what combination of distributions is chosen the schemes proposed by nobleet al and by inamuroet al yield the correct zeroslip velocity while some other schemes such as the bounce back scheme and the equilibrium distribution scheme would inevitably generate a nonzero slip velocity the bounce back scheme with the wall located halfway between a flow node and a bounce back node is also studied for the simple flows considered and is shown to produce results of second order accuracy the momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries to be specific the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids

finite difference lattice boltzmann methods for binary fluids we investigate two fluid bhatnagar gross krook bgk kinetic methods for binary fluids the developed theory works for asymmetric as well as symmetric systems for symmetric systems it recovers sirovich s theory and is summarized in models a and b for asymmetric systems it contributes models c d and e which are especially useful when the total masses and or local temperatures of the two components are greatly different the kinetic models are discretized based on an octagonal discrete velocity model the discrete velocity kinetic models and the continuous ones are required to describe the same hydrodynamic equations the combination of a discrete velocity kinetic model and an appropriate finite difference scheme composes a finite difference lattice boltzmann method the validity of the formulated methods is verified by investigating i uniform relaxation processes ii isothermal couette flow and iii diffusion behavior

on quadrature of highly oscillatory integrals with logarithmic singularities in this paper a quadrature rule is discussed for highly oscillatory integrals with logarithmic singularities at the same time its error depends on the frequency and the computation of its moments are given the new rule is implemented by interpolating f at chebyshev nodes and singular point where the interpolation polynomial satisfies some conditions numerical experiments conform the efficiency for obtaining the approximations

a method for near equilibrium discrete velocity gas flows we present a simulation scheme for discrete velocity gases based on local thermodynamic equilibrium exploting the kinetic nature of discrete velocity gases in that context results in a natural splitting of fluxes and the resultant scheme strongly resembles the original process the kinetic nature of the scheme and the modeling of the infinite collision rate limit result in a small value of the coefficient of numerical viscosity the behavior of which is remarkably physical a first order method and two second order methods using the total variation diminishing principle are developed and an example application is presented given the same computer resources it is expected that with this approach a much higher reynold s number will be achievable than presently possible with either lattice gas automata or lattice boltzmann approaches the ideas being general the scheme is applicable to any discrete velocity model and to lattice gases as well

steady state convergence acceleration of the generalized lattice boltzmann equation with forcing term through preconditioning several applications exist in which lattice boltzmann methods lbm are used to compute stationary states of fluid motions particularly those driven or modulated by external forces standard lbm being explicit time marching in nature requires a long time to attain steady state convergence particularly at low mach numbers due to the disparity in characteristic speeds of propagation of different quantities in this paper we present a preconditioned generalized lattice boltzmann equation glbe with forcing term to accelerate steady state convergence to flows driven by external forces the use of multiple relaxation times in the glbe allows enhancement of the numerical stability particular focus is given in preconditioning external forces which can be spatially and temporally dependent in particular correct forms of moment projections of source forcing terms are derived such that they recover preconditioned navier stokes equations with non uniform external forces as an illustration we solve an extended system with a preconditioned lattice kinetic equation for magnetic induction field at low magnetic prandtl numbers which imposes lorentz forces on the flow of conducting fluids computational studies particularly in three dimensions for canonical problems show that the number of time steps needed to reach steady state is reduced by orders of magnitude with preconditioning in addition the preconditioning approach resulted in significantly improved stability characteristics when compared with the corresponding single relaxation time formulation

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

an error embedded method based on generalized chebyshev polynomials in this paper we develop an error embedded method based on generalized chebyshev polynomials for solving stiff initial value problems the solution and the error at each integration step are calculated by generalized chebyshev polynomials of two consecutive degrees having overlapping zeros which enables us to minimize overall computational costs further the errors at each integration step are embedded in the algorithm itself in terms of concrete convergence and stability analysis the constructed algorithm turns out to have the 6th order convergence and an almost l stability we assess the proposed method with several numerical results showing that it uses larger time step sizes and is numerically more efficient

cut free gentzen calculus for multimodal ck this paper extends previous work on the modal logic ck as a reference system both proof theoretically and model theoretically for a correspondence theory of constructive modal logics first the fundamental nature of ck is discussed and compared with the intuitionistic modal logic ik which is traditionally taken to be the base line then it is shown that ck admits of a cut free gentzen sequent calculus g ck which has i a local interpretation in constructive kripke models and ii does not require explicit world labels finally the paper demonstrates how non classical modal logics such as ik cs4 cl or email protected s deontic system of 2 sequents arise as theories of ck presented both as special rules and as frame classes

indefinite integration of oscillatory functions by the chebyshev series expansion abstract an automatic quadrature scheme is presented for evaluating the indefinite integral of oscillatory function x 0 t e i t dt 0 x 1 of a given function t which is usually assumed to be smooth the function t is expanded in the chebyshev series to make an efficient evaluation of the indefinite integral combining the automatic quadrature method obtained and sidi s extrapolation method makes an effective quadrature scheme for oscillatory infinite integral a x cos x d x for which numerical examples are also presented

on quadrature for cauchy principal value integrals of oscillatory functions the authors develop an algorithm for the numerical evaluation of cauchy principal value integrals of oscillatory functions the method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a jacobi weight a numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results bounds of the error and of the amplification factor are also provided

numerical stability analysis of fdlbm we analyze the numerical stability of finite difference lattice boltzmann method fdlbm by means of von neumann stability analysis the stability boundary of the fdlbm depends on the bgk relaxation time the cfl number the mean flow velocity and the wavenumber as the bgk relaxation time is increased at constant cfl number the stability of the central difference lb scheme may not be ensured the limits of maximum stable velocity are obtained around 0 39 0 43 and 0 43 for the central difference for the explicit upwind difference and for the semi implicit upwind difference schemes respectively we derive artificial viscosities for every difference scheme and investigate their influence on numerical stability the requirements for artificial viscosity is consistent with the conditions derived from von neumann stability analysis this analysis elucidates that the upwind difference schemes are suitable for simulation of high reynolds number flows

efficient evaluation of oscillatory bessel hilbert transforms in this paper we study the asymptotics and evaluation of the oscillatory bessel hilbert transform 0 f x email protected n wx dx with 0 t the singularity of the hilbert transform is transferred to an individual oscillatory integral independent of f x for this singular integral we present two methods one is the combination of a filon type method and a complex integration method the other is the combination of a filon type method and an adaptive clenshaw curtis quadrature the remaining integral which is nonsingular can be well calculated with a combination of a filon type method and a gauss laguerre quadrature the efficiency and accuracy of the proposed methods are illustrated by numerical examples

computing cauchy principal value integrals using a standard adaptive quadrature we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals a b f x x 1 d x a b using a standard adaptive quadrature in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

roundoff errors in the problem of computing cauchy principal value integrals we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals mathrm p int a b f x x tau 1 dx a tau b using standard adaptive quadratures in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

two dimensional lattice boltzmann model for compressible flows with high mach number abstract in this paper we present an improved lattice boltzmann model for compressible navier stokes system with high mach number the model is composed of three components i the discrete velocity model by m watari and m tsutahara phys rev e 67 2003 036306 ii a modified lax wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included iii artificial viscosity the improved model is convenient to compromise the high accuracy and stability the included dispersion term can effectively reduce the numerical oscillation at discontinuity the added artificial viscosity helps the scheme to satisfy the von neumann stability condition shock tubes and shock reflections are used to validate the new scheme in our numerical tests the mach numbers are successfully increased up to 20 or higher the flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

notions of computation and monads abstract the calculus is considered a useful mathematical tool in the study of programming languages since programs can be identified with terms however if one goes further and uses conversion to prove equivalence of programs then a gross simplification is introduced programs are identified with total functions from values to values that may jeopardise the applicability of theoretical results in this paper we introduce calculi based on a categorical semantics for computations that provide a correct basis for proving equivalence of programs for a wide range of notions of computation

preconditioned lattice boltzmann method for steady flows in this paper we propose a preconditioned lattice boltzmann lb method for steady incompressible flows for steady flows the macroscopic equations derived from this lb model are equivalent to those from the standard lb model but with an improved eigenvalue system the proposed model can be viewed as an explicit solver for preconditioned compressible navier stokes equations linear stability analysis is performed and the results show that the stability of the model is the same as that of the standard lb model for low mach numbers the proposed model retains the structure of the standard lb model and hence possesses all the advantages numerical tests show that the convergence rate can be enhanced as much as an order of magnitude compared to the standard lattice boltzmann method the accuracy of the solutions is improved as well

logic of subtyping we introduce new modal logical calculi that describe subtyping properties of cartesian product and disjoint union type constructors as well as mutually recursive types defined using those type constructors basic logic of subtyping s extends classical propositional logic by two new binary modalities and an interpretation of s is a function that maps standard connectives into set theoretical operations intersection union and complement and modalities into cartesian product and disjoint union type constructors this allows s to capture many subtyping properties of the above type constructors we also consider logics s and s that incorporate into s mutually recursive types over arbitrary and well founded universes correspondingly the main results are completeness of the above three logics with respect to appropriate type universes in addition we prove cut elimination theorem for s and establish decidability of s and s

uniform approximations to cauchy principal value integrals of oscillatory functions this paper presents an interpolatory type integration rule for the numerical evaluation of cauchy principal value integrals of oscillatory integrands 1 1e i w xf x email protected where 1 t 1 for a given smooth function f x the proposed method is constructed by interpolating f x at practical chebyshev points and subtracting out the singularity a numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast fourier transform the validity of the method has been demonstrated by several numerical experiments

a framework for linear authorization logics abstract linear authorization logics lals are logics based on linear logic that can be used for modeling effect based authentication policies lals have been used in the context of the proof carrying authorization framework where formal proofs must be constructed in order for a principal to gain access to some resource elsewhere this paper investigates the complexity of the provability problem that is determining whether a formula is provable in a linear authorization logic we show that the multiplicative propositional fragment of lal is already undecidable in the presence of two principals on the other hand we also identify a first order fragment of lal for which provability is pspace complete finally we argue by example that the latter fragment is natural and can be used in practice

computation of integrals with oscillatory and singular integrands using chebyshev expansions we present a general method for computing oscillatory integrals of the form 1 1f x g x e i w xdx where f is sufficiently smooth on 1 1 w is a positive parameter and g is a product of singular factors of algebraic or logarithmic type based on a chebyshev expansion of f and the properties of chebyshev polynomials the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor e i w x furthermore due to numerically stable recurrence relations for the modified moments the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration numerical examples are provided to confirm our analysis

a new algorithm for cauchy principal value and hadamard finite part integrals an algorithm for the approximate evaluation of integrals defined by cauchy principal value or by hadamard finite part has been proposed the convergence of the procedure is proved the stability of the algorithm is considered and some numerical examples are given

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method