adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

resolution of the 2d navier stokes equations in velocity vorticity form by means of an influence matrix technique abstract an influence matrix technique is proposed to enforce both the continuity equation and the definition of the vorticity in the treatment of the 2d incompressible navier stokes equations it is shown and supported by numerical experiments that at each time step the divergence is actually equal to zero within machine accuracy the same result is obtained for the definition of the vorticity

saddlepoint approximation for student s t statistic with no moment conditions a saddlepoint approximation of the student s t statistic was derived by daniels and young biometrika 78 1991 169 179 under the very stringent exponential moment condition that requires that the underlying density function go down at least as fast as a normal density in the tails this is a severe restriction on the approximation s applicability in this paper we show that this strong exponential moment restriction can be completely dispensed with that is saddlepoint approximation of the student s t statistic remains valid without any moment condition this confirms the folklore that the student s t statistic is robust against outliers the saddlepoint approximation neat only provides a very accurate approximation for the student s t statistic but it also can be applied much more widely in statistical inference as a result saddlepoint approximations should always be used whenever possible some numerical work will be given to illustrate these points

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

a spectral solver for the navier stokes equations in the velocity vorticity formulation for flows with two nonperiodic directions a novel pseudospectral solution procedure for the navier stokes equations in the velocity vorticity formulation suitable to simulate flows with two nonperiodic directions is proposed an influence matrix method including a tau correction procedure has been employed to elicit an algorithmic substitute for thea priorilacking boundary conditions for the vorticity following o daube j comput phys 103 402 1992 the influence matrix is built by enforcing either the definition of the vorticity or the continuity equation at the boundary of the domain the influence matrix is nonsingular in both cases the order of the influence matrix is twice larger than one would expect from analogous solution methods based on finite differences or finite elements the spatial discretization is based on a 2d chebyshev expansion on anonstaggeredgrid of collocation points and the boundary conditions are imposed via the lanczos tau procedure the time marching scheme is adams bashforth for the advection term and crank nicolson for the viscous term the proposed scheme yields machine accurate divergence free flow fields and the definition of the vorticity is satisfied within machine accuracy

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

a divergence free chebyshev collocation procedure for incompressible flows with two non periodic directions abstract the enforcement of divergence free condition in the interior and on the boundaries of an incompressible flow with two non periodic directions is discussed for a chebyshev collocation formulation an influence matrix technique along with a correction methodology is used to satisfy the continuity equation everywhere in the domain details of implementing this procedure in a collocation method are presented an efficient solution procedure based on matrix diagonalization has been used to solve the resulting full matrices two test problems a flow in a driven square cavity and b fully developed laminar flow in a square duct subject to a three dimensional perturbation are studied run time statistics cpu memory mflops of the solution procedure are presented for representative grid sizes

self normalized large deviations let x x n n geq 1 be a sequence of independent and identically distributed random variables the classical cramer chernoff large deviation states that lim n to infty n 1 ln p sum i 1 n x i n geq x ln rho x if and only if the moment generating function of x is finite in a right neighborhood of zero this paper uses n p 1 p v n p n p 1 p sum i 1 n x i p 1 p p 1 as the normalizing constant to establish a self normalized large deviation without any moment conditions a self normalized moderate deviation that is the asymptotic probability of p s n v n p geq x n for x n o n p 1 p is also found for any x in the domain of attraction of a normal or stable law as a consequence a precise constant in the self normalized law of the iterated logarithm of griffin and kuelbs is obtained applications to the limit distribution of self normalized sums the asymptotic probability of the t statistic as well as to the erdos renyi shepp law of large numbers are also discussed

on optimal message vector length for block single parallel partition algorithm in a three dimensional adi solver it has long been recognized that many direct parallel tridiagonal solvers are only efficient for solving a single tridiagonal equation of large sizes and they become inefficient when naively used in a three dimensional adi solver in order to improve the parallel efficiency of an adi solver using a direct parallel solver we implement the single parallel partition spp algorithm in conjunction with message vectorization which aggregates several communication messages into one to reduce the communication costs the measured performances show that the longest allowable message vector length mvl is not necessarily the best choice to understand this observation and optimize the performance we propose an improved model that takes the cache effect into consideration the optimal mvl for achieving the best performance is shown to depend on number of processors and grid sizes similar dependence of the optimal mvl is also found for the popular block pipelined method

fourier spectral method on ensemble architectures the direct simulation of three dimensional unsteady fluid flows is performed on some of the most powerful parallel computers presently available connection machine 2 intel i860 and the ncube 2 a fourier pseudospectral method and second order timestepping scheme is used implementation performance analysis and experimental timings are given results of fluid flow simulations on meshes up to 2563 show sustained rates of 520 mflops on the 64 k node cm 2 484 mflops on the 1 k node ncube 2 and 103 mflops on the 32 node intel i860

the kinematic laplacian equation method a novel procedure for the navier stokes equations in the vorticity velocity formulation is presented the time evolution of the vorticity is solved as an ode problem on each node of the spatial discretization using at each step of the time discretization the spatial solution for the velocity field provided by a new pde expression called the kinematic laplacian equation kle this complete decoupling of the two variables in a vorticity in time velocity in space split algorithm reduces the number of unknowns to solve in the time integration process and also favors the use of advanced ode algorithms enhancing the efficiency and robustness of time integration the issue of the imposition of vorticity boundary conditions is addressed as well as the details of the implementation of the kle by isoparametric finite element discretization we shall see some validation results of the kle method applied to the classical case of a circular cylinder in impulsive started pure translational steady motion at several reynolds numbers in the range 5

level set based deformation methods for adaptive grids a new method for generating adaptive moving grids is formulated based on physical quantities level set functions are used to construct the adaptive grids which are solutions of the standard level set evolution equation with the cartesian coordinates as initial values the intersection points of the level sets of the evolving functions form a new grid at each time the velocity vector in the evolution equation is chosen according to a monitor function and is equal to the node velocity a uniform grid is then deformed to a moving grid with desired cell volume distribution at each time the method achieves precise control over the jacobian determinant of the grid mapping as the traditional deformation method does the new method is consistent with the level set approach to dynamic moving interface problems

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

an almost sure central limit theorem for self normalized partial sums let x i i 1 be a sequence of i i d random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance denote s email protected i 1 nx i v n email protected i 1 nx i 2 then an almost sure central limit theorem for self normalized partial sums s n v n is studied under a mild condition in this paper

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

a vorticity velocity method for the numerical solution of 3d incompressible flows a new method for the numerical solution of the 3d navier stokes equations written in terms of vorticity velocity is presented the advantages of this formulation with respect to primitive variables and vorticity vector potential ones are discussed in view of physical as well as engineering applications a suitable form of the continuum equations the most appropriate discretization scheme and variable location in order to guarantee the solenoidality of the velocity and vorticity fields are introduced and justified a 3d lid driven cavity problem for 400 re 3200 is chosen as a test case for comparison and validation purposes a parallel implementation of the method as been performed on a shared memory architecture mainframe speedup results and efficiency considerations are given and discussed

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

a numerical study of the two dimensional navier stokes equations in vorticity velocity variables abstract solution methods for compact finite difference schemes are applied to the vorticity velocity form of the two dimensional unsteady navier stokes equations numerical experiments for stagnation point and driven cavity flows are described

direct numerical simulation of spatial transition to turbulence using fourth order vertical velocity second order vertical vorticity formulation a highly accurate algorithm has been developed to study the process of spatial transition to turbulence the algorithmic details of the direct numerical simulation dns of transition to turbulence in a boundary layer based on a formulation in terms of vertical velocity and vertical vorticity are presented issues concerning the boundary conditions are discussed the linear viscous terms are discretized using an implicit crank nicholson scheme and a low storage runge kutta method is used for the nonlinear terms for the spatial discretization fourth order compact finite differences have been used as these have been found to have better resolution compared to explicit differencing schemes of comparable order the number of grid points that are needed per wavelength is close to the theoretical optimum for any numerical scheme the resulting time discretized fourth order equations are split up into two second order equations resulting in helmholtz and poisson type equations the boundary conditions for the laplacian of the vertical velocity are determined using an influence matrix method a robust multigrid algorithm has been developed to solve the resulting anisotropic elliptical equations for the outflow boundary a buffer domain method which smoothly reduces the disturbances to zero in conjunction with parabolization of the navier stokes equations has been used the validation of the results for the dns solver is made both for linear and weakly nonlinear cases

finite element solution of the 3d compressible navier stokes equations by a velocity vorticity method this paper presents an alternative formulation to the primitive variable form of the navier stokes equations the approach is based on the use of the velocity and vorticity variables as a logical extension of the stream function vorticity method which is quite effective for two dimensional flows second order equations are obtained for the variables and discretization is through a weak galerkin finite element method followed by newton linearization the velocity and vorticity components are solved simultaneously by a direct solver the scheme is demonstrated for two dimensional and three dimensional incompressible and subsonic internal enclosed flow problems

the numerical solution of the navier stokes equations for 3 dimensional unsteady incompressible flows by compact schemes abstract this paper describes a numerical method for solving the navier stokes equations for unsteady incompressible 3 dimensional flows using velocity vorticity variables and irregular cartesian grids the method involves solving cauchy riemann type equations for the velocity and transport diffusion equations for the vorticity whose solenoidal vorticity components are obtained by solving a poisson equation for a suitably chosen scalar potential the importance of boundary conditions is examined the difference equations are solved by iterations in order to permit exploitation of parallel and vector computing methods numerical experiments confirm the second order spatial and temporal accuracy of the method

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

the treatment of spurious pressure modes in spectral incompressible flow calculations algorithms for the transient and steady state simulation of incompressible newtonian and non newtonian flows are described for the primitive variable formulation of the governing equations spectral approximations are used for the spatial discretization attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure spurious pressure modes are removed by means of a singular value decomposition the corresponding velocity field is divergence free at all the collocation points numerical results are presented for newtonian flow in a regularized driven square cavity and for non newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

when is the student t statistic asymptotically standard normal let x x i i in mathbb n be independent identically distributed random variables it is shown that the student t statistic based upon the sample x i i 1 n is asymptotically n 0 1 if and only if x is in the domain of attraction of the normal law it is also shown that for any x if the self normalized sums u n sum i 1 n x i sum i 1 n x i 2 1 2 n in mathbb n are stochastically bounded then they are uniformly subgaussian that is sup n mathbb e exp lambda u n 2 infty for some lambda 0

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

non linear characterizations for functions of hypergeometric type and their derivatives of any order abstract to find characterizations for special functions is an important question in applied mathematics and mathematical physics up to now most effort has been invested for polynomial sets which are orthogonal on the real line in this paper two non linear characterizations for a class of functions the so called functions of hypergeometric type are found not only for the functions themselves but also for their derivatives of any order members of this class are the bessel functions the hypergeometric functions the classical orthogonal polynomials as well as many other functions of mathematics and physics applications to the subclass of polynomials of hypergeometric type are made with emphasis on the classical orthogonal polynomials for which the coefficients of the above mentioned non linear characterizations are tabulated

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

on the advantages of the vorticity velocity formulations of the equations of fluid dynamics the mathematical properties of the pressure velocity and vorticity velocity formulations of the equations of viscous flow are compared it is shown that a vorticity velocity formulation exists which has the interesting property that noninertial effects only enter into the problem through the implementation of initial and boundary conditions this valuable characteristic along with other advantages of the vorticity velocity approach are discussed in detail

an adaptive mesh redistribution method for the incompressible mixture flows using phase field model a phase field model which describes the motion of mixtures of two incompressible fluids is presented by liu and shen c liu j shen a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method phys d 179 2003 211 228 the model is based on an energetic variational formulation in this work we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids it is a coupled nonlinear system of navier stokes equations and allen cahn phase equation phase field equation through an extra stress term and the transport term the numerical strategy is based on the approach proposed by li et al r li t tang p w zhang moving mesh methods in multiple dimensions based on harmonic maps j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution in the pde evolution part the phase field equation is numerically solved by a conservative scheme with a lagrange multiplier and the coupled incompressible navier stokes equations are solved by the incremental pressure correction projection scheme based on the semi staggered grid method in the mesh moving part the mesh points are iteratively redistributed by solving the euler lagrange equations with a parameter free monitor function in each iteration the pressure and the phase are updated on the resulting new grid by a conservative interpolation formula while the velocity is re mapped in a non conservative approach a simple method for preserving divergence free is obtained by projecting the velocity onto the divergence free space after generating the new mesh at the last iterative step numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows

divergence free velocity fields in nonperiodic geometries the influence matrix method of enforcing incompressibility in pseudospectral simulations of fluid dynamics as described by kleiser and schumann for channel flow is generalized to other geometries a formalism of projection and matrix operators is introduced in which the influence matrix method is shown to be an application of the classic sherman morrison woodbury formula of numerical linear algebra special attention is paid to the tau correction applications to cartesian geometries illustrate the concepts and highlight the role of symmetry a coded implementation in a cylindrical geometry requiring special treatment of coordinate singularities is used to investigate properties of the influence matrix and to provide estimates of timings

incompressibility and no slip boundaries in the chebyshev tau approximation i present a minor but important correction to kleiser and schumann s influence matrix solution for numerical simulation of 3d time dependent incompressible flow between no slip plane boundaries l kleiser and u schumann in proc 3rd gamm conf numerical methods in fluid mechanics edited by e h hirschel vieweg braunschweig 1980 p 165 the solution technique i present is for the most part identical to kleiser and schumann s the only noteworthy difference being my treatment of the truncation or tau errors though this improvement appears to be but a slight modification to their original algorithm the error encountered when attempting to satisfy the incompressibility constraint is reduced dramatically by a factor of 1011 producing velocity fields which are divergence free to within machine precision

donsker s theorem for self normalized partial sums processes let x x1 x2 be a sequence of nondegenerate i i d random variables with zero means in this paper we show that a self normalized version of donsker s theorem holds only under the assumption that x belongs to the domain of attraction of the normal law a thus resulting extension of the arc sine law is also discussed we also establish that a weak invariance principle holds true for self normalized self randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero if and only if max1 j n xj vn p0 as n where v2n nj 1x2j

hopf bifurcation of the unsteady regularized driven cavity flow abstract a numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a chebyshev tau approximation for the space variables the high accuracy of the spectral methods and the condensed distribution of the chebyshev collocation points near the boundary enable us to obtain reliable results for high reynolds numbers with a moderate number of modes it is found that the flow converges to a stationary state for reynolds numbers re up to 10 000 for reynolds numbers larger than a critical value 10 000

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

tail probability approximations for student s t statistics in this paper we derive saddlepoint approximations for student s t statistics for strongly nonlattice random variables without moment conditions under very mild conditions we show that saddlepoint equations always have solutions

computation of natural convection in two dimensional cavities with chebyshev polynomials abstract a semi implicit spectral method for the computation of buoyancy induced two dimensional flow of a viscous incompressible fluid is presented the incompressible navier stokes equations expressed in terms of the primitive variables velocity and pressure and an additional equation for the temperature are considered all the variables are expanded in double truncated series of chebyshev polynomials the pressure distribution at the boundary of the computational domain is determined by means of an influence matrix technique in order to satisfy the incompressibility condition everywhere in the field the algorithm is applied to the classical problem of natural convection of a boussinesq fluid within a differentially heated vertical square cavity results show that for a prandtl number of 0 71 corresponding to air and up to rayleigh numbers of 106 very accurate solutions are obtained with a spatial resolution of 33 polynomials in each direction new interesting phenomena are shown for higher values of the rayleigh number

a chebyshev collocation algorithm for 2d non boussinesq convection abstract a chebyshev collocation algorithm is developed to integrate the time dependent navier stokes equations for natural convection flow with large temperature differences the working fluid is assumed to be a perfect gas and its thermophysical properties vary with temperature according to sutherland laws the governing equations do not allow for acoustic waves the generalized helmholtz and uzawa operators which arise from time discretization are solved iteratively and the performances of several types of preconditioners and iterative schemes are examined the algorithm is validated by computing almost boussinesq flows and by comparing with previous results obtained with a finite difference algorithm we investigate the effects of the temperature difference and of total mass contained within the cavity on the transition to unsteadiness in a cavity of aspect ratio 8 it is shown that these parameters have indeed a significant effect on the value of rayleigh number at which unsteadiness is triggered we also discuss the nature of the time periodic solution which is obtained for ra slightly supercritical values

arbitrary lagrangian eulerian finite element analysis of free surface flow using a velocity vorticity formulation this paper describes the application of velocity vorticity formulation of the navier stokes equations for two dimensional free surface flow using an arbitrary lagrangian eulerian method the velocity poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain the boundary fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method the numerical model for the velocity vorticity formulation is validated for a square cavity flow at re 400 and 1000 the solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model then the proposed numerical model is used to obtain flow results for the following free surface flow cases i interaction between two opposite solitary waves ii seiche phenomenon in a rectangular reservoir and iii solitary wave through a submerged rectangular structure in a viscous fluid the efficiency of the present numerical model for numerical treatment of free surface flows is discussed furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view

application of vorticity integral conditioning to chebyshev pseudospectral formulation for the navier stokes equations a pseudospectral method based on the vorticity stream function formulation is proposed for the solution of two dimensional time dependent flow of an incompressible fluid it features the high resolution and computational economy of chebyshev collocation and the combined second order adams bashforth and crank nicolson time integration schemes precise treatment for the vorticity condition is accomplished by the use of conditions of integral type numerical experiments indicate that the pseudospectral method is capable of producing results comparable to those obtained by finite differences with fewer unknowns and is superior in accuracy for the same number of nodal points

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

global spectral analysis of multi level time integration schemes an analysis is reported here for three time level integration methods following the global spectral analysis gsa described in high accuracy computing methods t k sengupta cambridge univ press usa the focus is on the second order adamsbashforth ab2 and the extrapolation in time ext2 methods careful distinction is made for the first time step at t 0 by either euler forward or four stage fourth order rungekutta rk4 time schemes the latter is used to solve a benchmark aeroacoustic problem several one dimensional wave propagation models are analyzed pure advection and advection diffusion equations various spatial discretizations are discussed including fourier spectral method attention is paid to the presence of physical and numerical modes as noted in the quadratic equation obtained from the difference equation for the model 1d convection equation it is shown that ab2 method is less stable and accurate than ext2 method with respect to numerical dissipation and dispersion this is true for the methods in which the physical mode dominates over the numerical mode presented analysis provides useful guide to analyze any three time level methods

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

self normalized laws of the iterated logarithm on demontre une loi du logarithme itere qui generalise la version classique pour toutes les distributions dans la classe de feller pour cette demonstration l auteur utilise une auto normalisation convenable pour les sommes partielles de variables aleatoires independantes et identiquement distribuees un cas particulier de ces resultats s applique a toute distribution dans le domaine de l attraction d une certaine loi stable

a compact monotonic discretization scheme for solving second order vorticity velocity equations abstract this paper presents a numerical method for solving the steady state navier stokes equations for incompressible fluid flows using velocities and vorticity as working variables the method involves solving a second order differential equation for the velocity and a convection diffusion equation for the vorticity in cartesian grids the key to the success of the numerical simulation of this class of flow equations depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity in this paper we present a monotonic advection diffusion multi dimensional scheme and a theoretically rigorous implementation of vorticity boundary conditions while the derivation of the proposed integral vorticity boundary condition is more elaborate and is more difficult to solve than conventional local approaches the present approach offers significant advantages in this study both lid driven and backward facing step problems have been selected for comparison and validation purposes

a new parallel strategy for two dimensional incompressible flow simulations using pseudo spectral methods a novel parallel technique for fourier galerkin pseudo spectral methods with applications to two dimensional navier stokes equations and inviscid boussinesq approximation equations is presented it takes advantage of the programming structure of the phase shift de aliased scheme for pseudo spectral codes and combines the task distribution strategy z yin h j h clercx d c montgomery an easily implemented task based parallel scheme for the fourier pseudo spectral solver applied to 2d navier stokes turbulence comput fluid 33 2004 509 and parallelized fast fourier transform scheme the performances of the resulting mpi fortran90 codes with the new procedure on sgi 3800 are reported for fixed resolution of the same problem the peak speed of the new scheme can be twice as fast as the old parallel methods the parallelized codes are used to solve some challenging numerical problems governed by the navier stokes equations and the boussinesq equations two interesting physical problems namely the double valued w j structure in two dimensional decaying turbulence and the collapse of the bubble cap in the boussinesq simulation are solved by using the proposed parallel algorithms

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

galerkin spectral method for the vorticity and stream function equations a galerkin legendre spectral method for the solution of the vorticity and stream function equations in uncoupled form under no slip conditions in a square domain is presented which fully exploits the separation of variables in the two elliptic problems benefits from a nonsingular influence matrix and is able to solve the singular driven cavity problem modulo gibbs phenomenon even without regularizing the boundary condition at the corners

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

optimal mass transport based adaptive mesh method for phase field models of two phase fluid flows in this paper we present an efficient adaptive mesh method for solving phase field model of a mixture of two phase incompressible fluid flows the adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation also known as mongekantorovich problem mkp the numerical solution of the mkp is computed by taking the gradient of the steady state solution of a parabolic mongeampre equation pmae several numerical experiments are conducted to demonstrate the performance of the pmae method for solving phase field models the numerical results illustrate that the adaptive mesh computed with pmae method captures well the moving interface of the phase field function

finite time singularities in ideal fluids with swirl three dimensional ideal incompressible fluids with swirl are studied numerically using two different methods standard finite differences and a projection method based on upwind differencing both methods give quantitatively similar results leading to the conclusion that singularities form in finite time in a manner consistent with known theoretical criteria the effect of singularities in incompressible flows on nearby compressible flows is discussed

fourier spectral and wavelet solvers for the incompressible navier stokes equations with volume penalization convergence of a dipole wall collision in this study we use volume penalization to mimic the presence of obstacles in a flow or a domain with no slip boundaries this allows in principle the use of fast fourier spectral methods and coherent vortex simulation techniques based on wavelet decomposition of the flow variables to compute turbulent wall bounded flow or flows around solid obstacles by simply adding one term in the equation convergence checks are reported using a recently revived and unexpectedly difficult dipole wall collision as a benchmark computation several quantities like the vorticity isolines truncation error kinetic energy and enstrophy are inspected for a collision of a dipole with a no slip wall and compared with available benchmark data obtained with a standard chebyshev pseudospectral method we quantify the possible deteriorating effects of the gibbs phenomenon present in the fourier based schemes due to continuity restrictions of the penalized navier stokes equations on the wall it is found that gibbs oscillations have a negligible effect on the flow evolution allowing higher order recovery of the accuracy on a fourier basis by means of postprocessing an advantage of coherent vortex simulations on the other hand is that the degrees of freedom of the flow computation can strongly be reduced in this study we quantify the possible reduction of degrees of freedom while keeping the accuracy for an optimal convergence scenario the penalization parameter has to scale with the number of fourier and wavelet modes in addition an implicit treatment of the darcy drag term in the penalized navier stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements

the parallel fourier pseudospectral method abstract parallel algorithms of the fourier pseudospectral method are presented for the solution of the unsteady incompressible navier stokes equations the only major operation that requires parallelization is the multidimensional fft in tests performed on a 1024 node hypercube computer an efficiency of about 83 is obtained for a three dimensional problem with mesh size 128 3 the all fortran code requires 17 s per timestep rivalling rates obtained from optimized codes on current supercomputers

a low cost parallel implementation of direct numerical simulation of wall turbulence a numerical method for the direct numerical simulation of incompressible wall turbulence in rectangular and cylindrical geometries is presented the distinctive feature resides in its design being targeted towards an efficient distributed memory parallel computing on commodity hardware the adopted discretization is spectral in the two homogeneous directions fourth order accurate compact finite difference schemes over a variable spacing mesh in the wall normal direction are key to our parallel implementation the parallel algorithm is designed in such a way as to minimize data exchange among the computing machines and in particular to avoid taking a global transpose of the data during the pseudo spectral evaluation of the non linear terms the computing machines can then be connected to each other through low cost network devices the code is optimized for memory requirements which can moreover be subdivided among the computing nodes the layout of a simple dedicated and optimized computing system based on commodity hardware is described the performance of the numerical method on this computing system is evaluated and compared with that of other codes described in the literature as well as with that of the same code implementing a commonly employed strategy for the pseudo spectral calculation

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

recurrence relations for radial wavefunctions for the nth dimensional oscillators and hydrogenlike atoms we present a general procedure for finding recurrence relations of the radial wavefunctions for nth dimensional isotropic harmonic oscillators and hydrogenlike atoms

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

chebyshev collocation solutions of the navier stokes equations using multi domain decomposition and finite element preconditioning abstract the steady navier stokes equations are solved using series of basis functions involving chebyshev polynomials the projection method is a collocation scheme a newton s linearization is performed in order to obtain a set of algebraic equations as the matrix system is ill conditioned the collocation technique is preconditioned by a standard galerkin finite element method using a 9 nodes lagrangian element which presents decisive advantages sparsity reduced condition number easy treatment of complicated geometries to handle nontrivial geometries in the collocation process a domain decomposition is set up the treatment of interface conditions is fully described several test problems like the regularized driven square cavity and the backward facing step are discussed to show the abilities of the present algorithm

a practical guide to direct optimization for planar grid generation abstract in this paper we provide a guide to using the direct optimization formulation of variational grid generation particular emphasis is placed on the smoothness or length functional this is undoubtedly the most important functional in variational grid generation producing smooth grids and ensuring well posedness of the minimization problem when combined with more unruly functionals unfortunately in its most primary form length can produce folded grids when used with nonconvex geometries historically there have been two solutions to this dilemma using an inverse mapping the famous winslow generator or augmenting the functional with others that promote unicity both strategies have the drawback that the resulting minimization problem becomes complicated and expensive as another alternative we introduce a generalized strategy for length which does not use inverse mappings or auxiliary functionals but makes strong use of reference grids this strategy provides flexibility in controlling grid quality and its minimization problems can be solved using a simple multigrid algorithm yielding a robust grid generation scheme with optimal complexity we also survey recent developments in the use of variational grid generation in the form of direct optimization for the alignment problem

general recurrence and ladder relations of hypergeometric type functions a method for the explicit construction of general linear sum rules involving hypergeometric type functions and their derivatives of any order is developed this method only requires the knowledge of the coefficients of the differential equation that they satisfy namely the hypergeometric type differential equation special attention is paid to the differential recurrence or ladder relations and to the fundamental three term recurrence formulas most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials are instances of these sum rules moreover an extension of the method to the generalized hypergeometric type functions is also described allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen like atoms where the varying parameter is the dimensionality

structure function scaling of bounded two dimensional turbulence statistical properties of forced two dimensional turbulence generated in two different flow domains are investigated by numerical simulations the considered geometries are the square domain and the periodic channel domain both bounded by lateral no slip sidewalls the focus is on the direct enstrophy cascade range and how the statistical properties change in the presence of no slip boundaries the scaling exponents of the velocity and the vorticity structure functions are compared to the classical kraichnan batchelor leith kbl theory which assumes isotropy homogeneity and self similarity for turbulence scales between the forcing and dissipation scale our investigation reveals that in the interior of the flow domain turbulence can be considered statistically isotropic and locally homogeneous for the enstrophy cascade range but it is weakly intermittent however the scaling of the vorticity structure function indicates a steeper slope for the energy spectrum than the kbl theory predicts near the walls the turbulence is strongly anisotropic at all flow scales