generalized bibasic hypergeometric series and theiru n extensions we survey summation theorems for generalized bibasic hypergeometric series found recently by chu and perfected by macdonald these contain arbitrary sequences of parameters and generalize bibasic summation theorems found by gosper gasper and rahman they also contain a host of matrix inversions for infinite lower triangular matrices including those found by gould hsu carlitz and krattenthaler further special cases of krattenthaler s theorem are andrews matrix formulation of the bailey transform and bressoud s and gasper s inversions we extend the telescoping arguments used for bibasic summation theorems to provideu n 1 extensions of gosper s and gasper s bibasic summation theorems special cases includeu n 1 extensions of carlitz s and gasper s matrix inversions further specializations correspond to theu n 1 bailey transform andu n 1 bressoud matrix inversion found previously by milne applications includeu n 1 transformation and expansion formulas and au n 1 extension of aq analogue of abel s binomial theorem thisq analogue is different from those found by jackson and johnson

inversion of the pieri formula for macdonald polynomials we give the explicit analytic development of macdonald polynomials in terms of modified complete and elementary symmetric functions these expansions are obtained by inverting the pieri formula specialization yields similar developments for monomial jack and hall littlewood symmetric functions

on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

spurious transients of projection methods in microflow simulations the temporal behavior of projection methods for viscous incompressible lowreynolds number ows is adressed the methods considered result from algebraically splitting the linear system corresponding to each time step in such a way that the computation of velocity is segregated from that of pressure each method is characterized by two possibly equal approximate inverses b1 and b2 of the momentum equation velocity matrix plus a parameter which renders the method non incremental if 0 or incremental if 1 the classical rst order projection method together with more sophisticated methods perot s second order method yosida method pseudoexact factorization method and their incremental variants are put into the same algebraic form and their accuracy numerically tested splitting errors of rst

analysis of an exact fractional step method an exact fractional step or projection method for solving the incompressible navier stokes equations is analyzed the method is applied to both structured and unstructured staggered mesh schemes there are no splitting errors associated with the method it satisfies the incompressibility condition to machine precision and reduces the number of unknowns the exact projection technique is demonstrated on a two dimensional cavity flow and a multiply connected moving domain with a free surface its performance is compared directly to classic fractional step methods and shown to be roughly twice as efficient boundary conditions and the relationship of the method to streamfunction vorticity methods are discussed

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

linear independence of compactly supported separable shearlet systems abstract this paper examines linear independence of shearlet systems this property has already been studied for wavelets and other systems such as for instance for gabor systems in fact for gabor systems this problem is commonly known as the hrt conjecture in this paper we present a proof of linear independence of compactly supported separable shearlet systems for this we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements

high order accurate solution of the incompressible navier stokes equations high order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation with this goal in mind the incompressible navier stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid the equations are integrated in time by a second order semi implicit method stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions the method is extended to three dimensions by a fourier expansion in every time step a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning the convergence properties of the iterative method are analyzed the order of accuracy of the method is demonstrated in numerical experiments the method is used to compute the flow in a channel the driven cavity and a constricted channel

decomposition numbers for finite coxeter groups and generalised non crossing partitions given a finite irreducible coxeter group w a positive integer d and types t 1 t 2 t d in the sense of the classification of finite coxeter groups we compute the number of decompositions c 1 2 d of a coxeter element c of w such that i is a coxeter element in a subgroup of type t i in w i 1 2 d and such that the factorisation is minimal in the sense that the sum of the ranks of the t i s i 1 2 d equals the rank of w for the exceptional types these decomposition numbers have been computed by the first author in tropics in discrete mathematics m klazar et al eds springer verlag berlin new york 2006 pp 93 126 and seminaire lotharingien combin 54 2006 article b541 the type a n decomposition numbers have been computed by goulden and jackson in europ j combin 13 1992 357 365 albeit using a somewhat different language we explain how to extract the type b n decomposition numbers from results of bona bousquet labelle and leroux adv appl math 24 2000 22 56 on map enumeration our formula for the type d n decomposition numbers is new these results are then used to determine for a fixed positive integer l and fixed integers r 1 r 2 r l the number of multi chains 1 2 l in armstrong s generalised non crossing partitions poset where the poset rank of i equals r i and where the block structure of 1 is prescribed we demonstrate that this result implies all known enumerative results on ordinary and generalised non crossing partitions via appropriate summations surprisingly this result on multi chain enumeration is new even for the original non crossing partitions of kreweras moreover the result allows one to solve the problem of rank selected chain enumeration in the type d n generalised non crossing partitions poset which in turn leads to a proof of armstrong s f m conjecture in type d n thus completing a computational proof of the f m conjecture for all types it also allows one to address another conjecture of armstrong on maximal intervals containing a random multi chain in the generalised non crossing partitions poset

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

an accurate second order approximation factorization method for time dependent incompressible navier stokes equations in spherical polar coordinates a finite difference method for solving three dimensional time dependent incompressible navier stokes equations in spherical polar coordinates is presented in detail a new algorithm which is second order accurate in time and space is considered and decoupling between the velocity and the pressure is achieved by this algorithm further the numerical method is tested by computing the spherical couette flow between two concentric spheres with the inner one rotating a comparison of the numerical solutions with available numerical results and experimental measurements is made it is demonstrated that the numerical code is valid for solving three dimensional unsteady incompressible navier stokes equations in spherical polar coordinates

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

linearized factorization techniques for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four approximate factorization techniques which do not account for the approximate factorization errors are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

approximate factorization as a high order splitting for the implicit incompressible flow equations abstract we apply the method of approximate factorization to derive a second order accurate splitting of the incompressible flow equations stokes or navier stokes equations this is novel because the method of approximate factorization was believed inapplicable to this type of equation system we demonstrate the resulting splitting on a second order crank nicolson discretization and point out its intimate relationship to some existing second order accurate splitting projection methods further we use the approximate factorization method to derive entirely new splittings we first develop a new generalized second order accurate splitting which may be specialized to a variety of applications we indicate its application to finite elements checker board free cell centered discretizations and ocean modeling we then generalized the original splitting to an arbitrarily high order scheme

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

efficient algorithm on a nonstaggered mesh for simulating rayleigh benard convection in a box an efficient semi implicit second order accurate finite difference method is described for studying incompressible rayleigh benard convection in a box with sidewalls that are periodic thermally insulated or thermally conducting operator splitting and a projection method reduce the algorithm at each time step to the solution of four helmholtz equations and one poisson equation and these are are solved by fast direct methods the method is numeri cally stable even though all field values are placed on a single non staggered mesh commensurate with the boundaries the efficiency and accuracy of the method are characterized for several representative convection problems

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

applications of q lagrange inversion to basic hypergeometric series a family of g lagrange inversion formulas is given special cases include quadratic and cubic transformations for basic hypergeometric series the g analogs of the so called strange evaluations are also corollaries some new rogers ramanujan identities are given a connection between the work of rogers and andrews and lagrange inversion is stated

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

on warnaar s elliptic matrix inversion and karlsson minton type elliptic hypergeometric series using krattenthaler s operator method we give a new proof of warnaar s recent elliptic extension of krattenthaler s matrix inversion further using a theta function identity closely related to warnaar s inversion we derive summation and transformation formulas for elliptic hypergeometric series of karlsson minton type a special case yields a particular summation that was used by warnaar to derive quadratic cubic and quartic transformations for elliptic hypergeometric series starting from another theta function identity we derive yet different summation and transformation formulas for elliptic hypergeometric series of karlsson minton type these latter identities seem quite unusual and appear to be new already in the trigonometric i e p 0 case

a novel extension of the lagrange burmann expansion formula abstract aking use of factorization and inverse of the leibniz functional matrix in yang generalized leibniz functional matrices and factorizations of some well known matrices linear algebra appl 430 2009 511 531 we establish a formal extension of the classical lagrange burmann expansion formula in connection with this result some new matrix inversions are also investigated

highly energy conservative finite difference method for the cylindrical coordinate system a highly energy conservative second order accurate finite difference method for the cylindrical coordinate system is developed it is rigorously proved that energy conservation in discretized space is satisfied when appropriate interpolation schemes are used this argument holds not only for an unequally spaced mesh but also for an equally spaced mesh on cylindrical coordinates but not on cartesian coordinates numerical tests are undertaken for an inviscid flow with various schemes and it turns out that the proposed scheme offers a superior energy conservation property and greater stability than the intuitive and previously proposed methods for both equally spaced and unequally spaced meshes

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

non commutative extensions of the macmahon master theorem we present several non commutative extensions of the macmahon master theorem further extending the results of cartier foata and garoufalidis le zeilberger the proofs are combinatorial and new even in the classical cases we also give applications to the s extension and krattenthaler schlosser s q analogue

decomposition of riesz frames and wavelets into a finite union of linearly independent sets we characterize riesz frames and prove that every riesz frame is the union of a finite number of riesz sequences furthermore it is shown that for piecewise continuous wavelets with compact support the associated regular wavelet systems can be decomposed into a finite number of linearly independent sets finally for finite sets an equivalent condition for decomposition into a given number of linearly independent sets is presented

implicit compact linearized θ methods with factorization for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit three point compact linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four compact approximate factorization techniques which do not account for the approximate factorization errors and which involve three point stencils for each one dimensional operator are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized compact implicit method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

a bridge between projection methods and simple type methods for incompressible navier stokes equations a bridge is built between projection methods and simple type methods semi implicit method for pressure linked equation a general second order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non linear update of pressure term as n pn 1 i n pn where n is a coefficient matrix which may depend on the grid size time step and even velocity it includes three and four step projection methods the standard simple method is written in a concise formula for steady and unsteady flows it is proven that simple type methods have second order temporal accuracy for unsteady flows the classical second order projection method and simple type methods are united within the framework of the general second order projection formula two iteration algorithms of simple type methods for unsteady flows are described and discussed in addition detailed formulae are provided for general projection methods by using the runge kutta technique to update the convective term and crank nicholson scheme for the diffusion term copyright 2007 john wiley sons ltd

operator methods and lagrange inversion a unified approach to lagrange formulas on presente une methode generale de demonstration des formules d inversion de lagrange et de leurs variantes on considere le cas de plusieurs variables

the discrete shearlet transform a new directional transform and compactly supported shearlet frames it is now widely acknowledged that analyzing the intrinsic geometrical features of the underlying image is essential in many applications including image processing in order to achieve this several directional image representation schemes have been proposed in this paper we develop the discrete shearlet transform dst which provides efficient multiscale directional representation and show that the implementation of the transform is built in the discrete framework based on a multiresolution analysis mra we assess the performance of the dst in image denoising and approximation applications in image approximations our approximation scheme using the dst outperforms the discrete wavelet transform dwt while the computational cost of our scheme is comparable to the dwt also in image denoising the dst compares favorably with other existing transforms in the literature

finite difference method for incompressible navier stokes equations in arbitrary orthogonal curvilinear coordinates a finite difference method for solving three dimensional time dependent incompressible navier stokes equations in arbitrary curvilinear orthogonal coordinates is presented the method is oriented on turbulent flow simulations and consists of a second order central difference approximation in space and a third order semi implicit runge kutta scheme for time advancement spatial discretization retains some important properties of the navier stokes equations including energy conservation by the nonlinear and pressure gradient terms numerical tests cover cartesian cylindrical polar spherical cylindrical elliptic and cylindrical bipolar coordinate systems both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method

a consistent direct discretization scheme on cartesian grids for convective and conjugate heat transfer a new discretization scheme on cartesian grids namely a consistent direct discretization scheme is proposed for solving incompressible flows with convective and conjugate heat transfer around a solid object the navier stokes and the pressure poisson equations are discretized directly even in the immediate vicinity of a solid boundary with the aid of the consistency between the face velocity and the pressure gradient from verifications in fundamental flow problems the present method is found to significantly improve the accuracy of the velocity and the wall shear stress it is also confirmed that the numerical results are less sensitive to the courant number owing to the consistency between the velocity and pressure fields the concept of the consistent direct discretization scheme is also explored for the thermal field the energy equations for the fluid and solid phases are discretized directly while satisfying the thermal relations that should be valid at their interface it takes different forms depending on the thermal boundary conditions dirichlet isothermal and neumann adiabatic iso heat flux boundary conditions for convective heat transfer and a fluid solid thermal interaction for conjugate heat transfer the validity of these discretizations is assessed by comparing the simulated results with analytical solutions for the respective thermal boundary conditions and it is confirmed that the present schemes also show high accuracy for the thermal field a significant improvement for the conjugate heat transfer problems is that the second order spatial accuracy and numerical stability are maintained even under severe conditions of near practical physical properties for the fluid and solid phases

uncoupled temporally second order accurate implicit solver of incompressible navier stokes equations an efficient method with temporal second order accuracy for solving the incompressible navier stokes equations in generalized coordinate systems is developed efficiency is obtained by extending an existing fractional step semi implicit solver into an implicit one with implicit approximation of the convection terms a novel three level linearization scheme is proposed to decouple the momentum equations from one another in addition to the fractional step approach that decouples the continuity equation from the momentum equations consequently all of the governing equations are uncoupled without degrading temporal accuracy or stability and without the need for iterations to solve the nonlinear system of equations at each time level the proposed decoupling technique can be used for other systems of nonlinear and time dependent partial differential equations moreover only minor modifications are required to apply the method to existing two level schemes several test cases confirm that the proposed decoupled scheme has temporal second order accuracy and allows the use of large time steps

a scalable consistent second order sph solver for unsteady low reynolds number flows abstract smoothed particle hydrodynamics sph has successfully been used to study a variety of cases involving nearly inviscid flows where conservation properties allow for good physical approximation despite poor theoretical approximation properties of differential operators when used to study unsteady low reynolds number flow with large dissipation conservation alone cannot ensure quality of approximation and the traditional approach is inconsistent an alternative formulation has recently become popular making use of an approximate splitting scheme to ensure a divergence free velocity field however this scheme relies on an inconsistent discretization of the laplacian that diverges as particles become disordered under flow we present an incremental pressure correction scheme and combination of existing differential operator renormalizations that are able to achieve second order accuracy in time and space a brief review of sph approximation theory is provided to highlight the necessity of these renormalizations in implementing an approximate factorization scheme we demonstrate that when fast algebraic multigrid preconditioners are used to solve the resulting linear systems the scheme results in a consistent approximation that is scalable and amenable to parallelization several validation cases are presented for which a speedup of several orders of magnitude is achieved over traditional sph approaches finally an interface has been developed between the particle library lammps and the sparse linear algebra libraries in trilinos providing a massively parallel 3d sph capability scaling results for up to 134 million particles on 32 768 cores are presented along with a demonstration of the capability to simulate complex 3d geometries these results show that the added complexity of applying the necessary consistency corrections actually provides a factor of four speed up per linear solver iteration versus the uncorrected case despite the additional cost of constructing the corrections the resulting library provides a method that is consistent efficient and second order in both space and time while maintaining the flexibility of classical sph for single phase flows

near wall les closure based on one dimensional turbulence modeling a novel near wall les closure model is developed based on a revised form of the one dimensional turbulence odt model of kerstein and is tested by performing les calculations of turbulent channel flow at reynolds numbers based on friction velocity ranging from 395 to 10 000 in contrast to previous models which invoke reynolds averaging near wall velocity fluctuations and turbulent transport are simulated down to the smallest scales and can be compared directly to dns data thus the approach provides more than just a boundary condition rather it is itself a complete although simplified model for the wall normal profiles of velocity within the near wall region les odt coupling is bi directional and occurs both through the direct calculation of the subgrid turbulent stress by temporally and spatially filtering the odt resolved momentum fluxes up scale coupling and through the les resolved pressure and velocities impacting the odt behavior down scale coupling the formulation involves finely resolved odt lines that are embedded within each wall adjacent les cell denoted the inner region les cells that are within approximately one les filter width of the inner region belong to an overlap region where both odt and les modeling is active all other cells are treated using a standard les approach although more expensive than simpler models the cost of the model relative to the les portion of the simulation scales favorably with problem size leading to computationally affordable simulations even at relatively high reynolds numbers

a segregated implicit pressure projection method for incompressible flows in this paper a segregated implicit projection method is presented and investigated although high reynolds turbulent flow has been a motivating factor the present study is mainly restricted to the incompressible navier stokes equations the proposed method is analyzed for properties like stability accuracy and consistency aided by some test examples in addition two different flow cases are simulated to evaluate the ability and efficiency of the method compared with other results

time step restrictions using semi explicit methods for the incompressible navier stokes equations time step restrictions using semi explicit methods for the incompressible navier stokes equations