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

discontinuity capturing operators for elastodynamics finite element methods are presented which include discontinuity capturing operators to accurately solve elastodynamics problems exhibiting sharp gradients in their solution e g wave propagation the formulation involves finite element discretization of the temporal domain as well as the usual finite element discretization of the spatial domain linear least squares terms are included that enhance the stability of the space time finite element method nonlinear discontinuity capturing operators are presented that result in more accurate capturing of wave fronts in transient solutions while maintaining the high order accuracy of the underlying linear algorithm in smooth regions stability of the discontinuity capturing formulation is proved and a convergence proof is given for a particular choice of the discontinuity capturing operator numerical results are presented that demonstrate the effectiveness of the proposed discontinuity capturing operators for elastodynamics

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

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

some recent trends and developments in finite element analysis for incompressible thermal flows a numerical procedure for solving the time dependent incompressible navier stokes and energy equations is presented the present method is based on a set of finite element equations of the primitive variable formulation and a time splitting procedure which has unique features in its formulation as well as in its evaluation from the viewpoint of efficiency and accuracy the applicability of the proposed formulation was verified by computations and comparison with known results

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

algebraic fractional step schemes with spectral methods for the incompressible navier stokes equations the numerical investigation of a recent family of algebraic fractional step methods for the solution of the incompressible time dependent navier stokes equations is presented these methods are improved versions of the yosida method proposed in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations comput methods appl mech engrg 188 1 3 2000 505 526 a quarteroni f saleri a veneziani j math pures appl 9 78 5 1999 473 503 and one of them the yosida4 method is proposed in this paper for the first time they rely on an approximate lu block factorization of the matrix obtained after the discretization in time and space of the navier stokes system yielding a splitting in the velocity and pressure computation in this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method with the aim of investigating the impact of the splitting on the global accuracy of the computation

regular article a semi implicit numerical scheme for reacting flow a stiff 1 operator split projection scheme is constructed for the simulation of unsteady two dimensional reacting flow with detailed kinetics the scheme is based on the compressible conservation equations for an ideal gas mixture in the zero mach number limit the equations of motion are spatially discretized using second order centered differences and are advanced in time using a new stiff predictor corrector approach the new scheme is a modified version of the additive stiff scheme introduced in a previous effort by h n najm p s wyckoff and o m knio 1998 j comput phys 143 381 the predictor updates the scalar fields using a strang type operator split integration step which combines several explicit diffusion sub steps with a single stiff step for the reaction terms such that the global time step may significantly exceed the critical diffusion stability limit convection terms are explicitly handled using a second order multi step scheme the velocity field is advanced using a projection scheme which consists of a partial convection diffusion update followed by a pressure correction step a split approach is also adopted for the convection diffusion step in the momentum update this splitting combines an explicit treatment of the convective terms at the global time step with several explicit fractional steps for diffusion finally a corrector step is implemented in order to couple the evolution of the density and velocity fields and to stabilize the computations the corrector acts only on the convective terms and the pressure field while the predicted updates due to diffusion and reaction are left unchanged the correction of the scalar fields is implemented using a single step non split non stiff second order time integration a similar procedure is used for the velocity field which is followed by a pressure projection step the performance and behavior of the operator split scheme are first analyzed based on tests for a nonlinear reaction diffusion equation in one space dimension followed by computations with a detailed c 1c 2 methane air mechanism in one and two dimensions the tests are used to verify that the scheme is effectively second order in time and to suggest guidelines for selecting integration parameters including the number of fractional diffusion steps and tolerance levels in the stiff integration for two dimensional simulations with the present reaction mechanism flame conditions and resolution parameters speedup factors of about 5 are achieved over the previous additive scheme and about 25 over the original explicit scheme

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

a fast computation technique for the direct numerical simulation of rigid particulate flows in this paper we present a computation technique for the direct numerical simulation of freely moving rigid bodies in fluids we solve three dimensional laminar flow problems using a control volume approach the key feature of this approach is that the computational overhead relative to a pure fluid solver to solve for the motion of rigid particle is very small the formulation is convenient for handling irregular geometries we present results for the sedimentation of particles of different shapes convergence tests are presented to assess the order of accuracy of the numerical scheme various test cases are considered and the numerical results are compared with experimental values to validate the code due to the ability to perform fast computations this method has been used for animations and its application to the direct numerical simulation of turbulent particulate flows merits investigation the technique is not restricted to any constitutive model of the suspending fluid hence it may potentially be used in large eddy simulations les or reynolds averaged navier stokes rans type simulations

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

a divergence free condition compensated method for incompressible navier stokes equations the present study aims to develop a new formulation to effectively calculate the incompressible navier stokes solutions in non staggered grids the distinguished feature of the proposed method which avoids directly solving the divergence free equation is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility for the sake of numerical accuracy dispersion relation preserving upwind scheme developed within the two dimensional context was employed to approximate the convection terms the validity of the proposed mass preserving navier stokes method is justified by solving two benchmark problems at high reynolds and rayleigh numbers based on the simulated navier stokes solutions the proposed formulation is shown to outperform the conventional segregated method in terms of the reduction of cpu time 2007 elsevier b v all rights reserved

spectral tau approximation of the two dimensional stokes problem we analyse the spectral tau method for the approximation of the stokes system on a square with dirichlet boundary conditions we provide an error estimate in the norm of the sobolev spaceh s for the approximation of a divergence free vector field with polynomial divergence free vector fields we apply this result to prove some convergence estimates for the solution of the discrete stokes problem

a new family of projection schemes for the incompressible navier stokes equations with control of high frequency damping abstract a simple spatially discrete model problem consisting of mass points and dash pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible navier stokes equations in particular the temporal accuracy the stability and the numerical damping are investigated the present study suggests that it is not possible to formulate a second order accurate projection pressure correction scheme which possesses any high frequency damping motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised method respectively and offer control over high frequency damping both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible navier stokes equations comprehensive numerical studies of the flow in a lid driven cavity and the flow around a cylinder are presented the observations made are in agreement with the conclusions drawn from the model problem

a massively parallel fractional step solver for incompressible flows this paper presents a parallel implementation of fractional solvers for the incompressible navier stokes equations using an algebraic approach under this framework predictor corrector and incremental projection schemes are seen as sub classes of the same class making apparent its differences and similarities an additional advantage of this approach is to set a common basis for a parallelization strategy which can be extended to other split techniques or to compressible flows the predictor corrector scheme consists in solving the momentum equation and a modified continuity equation namely a simple iteration for the pressure schur complement consecutively in order to converge to the monolithic solution thus avoiding fractional errors on the other hand the incremental projection scheme solves only one iteration of the predictor corrector per time step and adds a correction equation to fulfill the mass conservation as shown in the paper these two schemes are very well suited for massively parallel implementation in fact when compared with monolithic schemes simpler solvers and preconditioners can be used to solve the non symmetric momentum equations gmres bi cgstab and to solve the symmetric continuity equation cg deflated cg this gives good speedup properties of the algorithm the implementation of the mesh partitioning technique is presented as well as the parallel performances and speedups for thousands of processors

how to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties abstract for a fairly general class of second order accurate algorithms a sub stepping procedure is described which achieves fourth order accuracy while retaining the stability conservation properties and implementation of the underlying second order method two features render the proposed approach especially attractive from a computational perspective over conventional fourth order methods first the improved fourth order accuracy is attained merely by two additional residual evaluations with no changes in the implementation of the underlying second order method in particular no additional storage or extra computations of high order gradients are required second for hamiltonian systems the stability properties and possible conservation properties of the underlying second order method such as energy momentum and symplectic character are preserved the new technique is applied to a number of standard algorithms for elastodynamics structural and rigid body dynamics in particular when applied to the recently proposed second order accurate energy momentum method this approach results in a fourth order accurate energy linear and angular momentum conserving algorithm for general hamiltonian systems

stable finite element solution of the incompressible navier stokes equations using linear interpolations for velocity and pressure abstract a velocity decomposition least squares method is reexamined and the results of several test cases are presented all the present calculations are based on linear triangular elements with a new treatment of boundary conditions for high reynolds number flows second order upwind schemes are adopted and the lanczos method is applied to accelerate the convergence of the iterative scheme which is based on the symmetric gauss seidel algorithm extension to unsteady flows is also presented

hypam a hybrid continuum particle model for incompressible free surface flows three major issues associated with numerical simulations of complex free surface flows viz interface tracking fragmentation and large physical jumps are addressed by a new hybrid continuum particle model hypam the new model consists of three parts 1 the polygonal area mapping method q zhang p l f liu a new interface tracking method the polygonal area mapping method j comput phys 227 8 2008 4063 4088 2 a new algorithm that decomposes the interested water phase into a continuum zone a buffer zone and a particle zone based on material topology and graph theory 3 a passive response assumption in which the air phase is assumed to respond passively to the continuum part of the water phase the incompressible inviscid euler equations and the equations describing the free fall of rigid bodies are used as the governing equations for the continuum buffer zone and the particle zone respectively and separately a number of examples including water droplet impact solitary wave propagation and dam break problems are simulated for the illustration and validation of hypam it is shown that hypam is more accurate and versatile than a continuum based volume of fluid model one major contribution of this work is the single phase decomposition algorithm useful for many other hybrid formulations neglecting surface tension viscosity and particle interactions hypam is currently limited to mildly fragmented free surface flows with high reynolds and weber numbers

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

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 stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

a well conditioned augmented system for solving navier stokes equations in irregular domains an augmented method based on a cartesian grid is proposed for the incompressible navier stokes equations in irregular domains the irregular domain is embedded into a rectangular one so that a fast poisson solver can be utilized in the projection method unlike several methods suggested in the literature that set the force strengths as unknowns which often results in an ill conditioned linear system we set the jump in the normal derivative of the velocity as the augmented variable the new approach improves the condition number of the system for the augmented variable significantly using the immersed interface method we are able to achieve second order accuracy for the velocity numerical results and comparisons to benchmark tests are given to validate the new method a lid driven cavity flow with multiple obstacles and different geometries are also presented

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

a finite volume formulation for transient convection and diffusion equations with unstructured distorted grids and its applications in fluid flow simulations with a collocated variable arrangement abstract this paper describes a finite volume method for simulating transport processes governed by convection diffusion type equations the formulation is based on a cell centred unstructured grid with an edge based data structure discretisation is independent of control volume or cell shape by using a surface vector decomposition at the midpoint of the interface between cells along with a deferred correction approach any cross diffusion due to grid skewness can be readily accounted for when discretising the diffusive flux for modelling fluid flow processes a collocated arrangement of variables is employed so that a single coefficient matrix applies for the momentum equations of each velocity component to avoid checkerboard oscillation arising from pressure velocity decoupling occurring under the collocated variable arrangement when a pressure based solution algorithm is employed a novel pseudo flux interpolation method is proposed for unstructured grids ensuring that the solution is both under relaxation factor and time step for transient calculation independent the methodology can be formulated within a framework whereby either a coupled or a decoupled solution algorithm can be employed the features and advantages of the method are demonstrated by solution of the navier stokes equations for two benchmark flow problems

a semi implicit numerical scheme for reacting flow an additive semi implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed the scheme relies on a zero mach number formulation of the compressible conservation equations with detailed chemistry the governing equations are discretized in space using second order differences and integrated in time using a semi implicit approach time integration of the evolution equations for species mass fraction thermodynamic pressure and density is performed using a semi implicit nonsplit scheme that combines a second order predictor corrector treatment of convection and diffusion terms and a stiff integrator for the reaction source terms meanwhile the momentum equations are integrated using a second order projection scheme the projection scheme is based on a predictor corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations a pressure poisson equation is inverted following both the predictor and corrector steps using a fast solver the advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion the comparison in based on both one dimensional 1d unsteady tests of a premixed methane air flame and unsteady two dimensional tests of the same flame interacting with a counterrotating vortex pair in both cases the grimech1 2 reaction mechanism with 32 species and 177 elementary reactions is used computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations for the present reaction mechanism and flame conditions speedup factors of about 10 are achieved in the 1d tests and about five in two dimensions possible extensions of the present scheme to further improve efficiency are also discussed

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

on some classes of nonstationary axially symmetric solutions to the navier stokes equations in the paper the navier stokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids the problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t the three functions are sought in the form of power series in r with coefficients depending on t and z for the unknown coefficients recurrence relations are obtained which contain three arbitrary functions the relations are examined in three particular cases in which they give analytical solutions to the navier stokes equations for any values of the coordinates t z and r

a high order semi implicit imex weno scheme for the all mach isentropic euler system abstract in this paper new high order schemes are constructed and analyzed for the numerical solution of euler equations of isentropic gas dynamics material waves are treated explicitly while acoustic waves are treated implicitly thus avoiding severe cfl restrictions for low mach flows high order accuracy in space is obtained by finite difference weno schemes while high order in time is obtained by imex methods with semi implicit linearization treatment the schemes are proven to be asymptotic preserving and asymptotic accurate as the mach number vanishes several tests in one and two space dimensions illustrate the effectiveness of the proposed schemes

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a second order accurate projection method for the incompressible navier stokes equations on non graded adaptive grids we present an unconditionally stable second order accurate projection method for the incompressible navier stokes equations on non graded adaptive cartesian grids we employ quadtree and octree data structures as an efficient means to represent the grid we use the supra convergent poisson solver of c h min f gibou h ceniceros a supra convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids cam report 05 29 j comput phys in press a second order accurate semi lagrangian method to update the momentum equation an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non graded grids we sample all the variables at the grid nodes producing a scheme that is straightforward to implement we propose two and three dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the l1 and l norms

on the efficiency of a numerical method with periodic time strides for solving incompressible flows an explicit numerical method to solve the unsteady incompressible flow equations consisting on n small time steps t between each two much larger time steps t 1 is considered the stability and efficiency of the method is first analyzed using the one dimensional diffusion equationl it is shown that the use of a time step t slightly smaller than the critical one t c given by numerical stability allows lo periodically take a much larger time step stride that speeds up the advance in time in a numerical stable scheme in particular the stability analysis shows that for a given value of the stride t 1 there is an optimum value of the small time step for which the computational speed is the fastest n is a minimum being this speed significantly larger than the corresponding one for an explicit method using t c only the efficiency of the method is discussed for different time discretization schemes the numerical method is then used to solve a particular incompressible flow it is shown that the method is significantly about three times faster than a standard explicit scheme and yields the same time evolution of the flow within spatial accuracy further it is shown that a much more higher computational speed and efficiency is reached if one combines an implicit scheme for the periodic strides with the explicit small time steps with this combination one can speed up the computations in more than one order of magnitude with the same resolution

a semi implicit method for internal boundary layers in compressible flows abstract a semi implicit boundary layer method is developed for application to spatially evolving internal boundary layers in compressible fluids the method circumvents convergence difficulties that may arise with fully implicit marching schemes whenever the mean streamwise velocity profile is inflectional the method is demonstrated for the special case of the compressible laminar axisymmetric jet however the properties of the scheme make it well suited also to laminar or turbulent free shear layers wakes and planar jets

explicit predictor multicorrector time discontinuous galerkin methods for non linear dynamics abstract explicit predictor multicorrector time discontinuous galerkin tdg methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non linear analysis of structural dynamics problems the formulation is intended to inherit the accuracy properties of the exact parent implicit tdg methods to this end suitable predictors and correctors are designed to achieve third order accuracy large stability limits and controllable numerical dissipation by means of an algorithmic parameter as the study of a general non linear case is rather complex the analysis of the convergence properties of the resulting algorithms are restricted to conservative duffing oscillators for which closed form solutions are available it is shown that the main properties of the underlying parent scheme can be retained finally results of representative numerical simulations relevant to duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates

higher entropy conservation and numerical stability of compressible turbulence simulations we present a numerical formulation for the treatment of nonlinear instabilities in shock free compressible turbulence simulations the formulation is high order and contains no artificial dissipation numerical stability is enhanced through semi discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation the numerical implementation is achieved using a conservative skew symmetric splitting of the nonlinear terms the robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high reynolds number results show the scheme is capable of capturing the statistical equilibrium of low mach number compressible turbulent fluctuations at infinite reynolds number comparisons with the entropy splitting technique j comput phys 162 2000 33 j comput phys 178 2002 307 staggered method j comput phys 191 2 2003 392 and skew symmetric like schemes j comput phys 161 2000 114 confirm the superiority of the current approach we also discuss a flaw in the skew symmetric splitting implemented in the literature very good results are obtained based on the proper splitting

a poly region boundary element method for incompressible viscous fluid flows a boundary element method bem for steady viscous fluid flow at high reynolds numbers is presented the new integral formulation with a poly region approach involves the use of the convective kernel with slight compressibility that was previously employed by grigoriev and fafurin 1 for driven cavity flows with reynolds numbers up to 1000 in order to avoid the overdeterminancy of the global set of equations when using eight noded rectangular volume cells from that previous work 12 noded hexagonal volume regions are introduced as a result the number of linearly independent integral equations for each node becomes equal to the degrees of freedom of the node the numerical results for square driven cavity flow having reynolds numbers up to 5000 are compared to those obtained by ghia et al 2 and demonstrate a high level of accuracy even in resolving the secondary vortices at the corners of the cavity next a comprehensive study is done for backward facing step flows at re 500 and 800 using the bem along with a standard galerkin based finite element methods fem the numerical methods are in excellent agreement with the benchmark solution published by gartling 3 however several additional aspects of the problem are also considered including the effect of the inflow boundary location and the traction singularity at the step corner furthermore a preliminary comparative study of the poly region bem versus the standard fem indicates that the new method is more than competitive in terms of accuracy and efficiency copyright 1999 john wiley sons ltd

estimation of the parameters of the rice distribution this paper deals with the problem of estimating the parameters of the rice distribution the distribution has applications in sonar and radar signal processing and a proper estimation procedure with associated confidence intervals is important using the sample second moment as an estimate of the second moment of the distribution two techniques viz methods of moments and maximum likelihood are applied to synthetic envelope data of known signal to noise ratios in order to estimate the parameter from different sample sizes it is concluded that the sample second moment is an unbiased estimate of the theoretical second moment and for the signal to noise ratio parameter both methods work without any significant bias and satisfy the criterion of maximum efficiency however the method of moments is simpler easier to apply and therefore recommended as the method of choice

adaptive finite element procedures for linear and non linear dynamics this paper discusses implementation and adaptivity of the discontinuous galerkin dg finite element method as applied to linear and non linear structural dynamic problems by the dg method both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived they are third order accurate and while the implicit procedure is unconditionally stable the explicit one is conditionally stable an h adaptive procedure based on the zienkiewicz zhu error estimate using the spr technique is applied numerical examples are presented to show the suitability of the dg method for both linear and non linear structural dynamic analysis copyright 1999 john wiley sons ltd

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

statistical 3 d beamforming for large scale mimo downlink systems over rician fading channels in this paper we investigate a downlink transmission algorithm for single cell multiuser systems with two dimensional 2 d large scale antenna array at the base station bs over rician fading channels we first derive some properties of the channel s line of sight los component in the large scale antenna array scenario next based on these properties and under the assumption of only statistical channel state information csi i e the los component and rician k factor at the bs we derive the optimal beamforming vector for each user in maximizing an approximation of the ergodic sum rate the main guidelines for user scheduling are also presented then a three dimensional 3 d beamforming downlink transmission algorithm exploiting only the statistical csi of each user is proposed for this algorithm we derive an exact analytical closed form expression for the achievable ergodic rate and present tractable approximations based on our analysis we gain some valuable insights the proposed algorithm is shown to perform well in achieving considerable sum rate while requiring much less csi at the bs it requires three scalar values at the bs for each user and can achieve an ergodic sum rate closer to the ergodic sum rate achieved by the matched filter precoding with perfect csi at the bs

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

deflated preconditioned conjugate gradient solvers for the pressure poisson equation a deflated preconditioned conjugate gradient technique has been developed for the solution of the pressure poisson equation within an incompressible flow solver the deflation is done using a region based decomposition of the unknowns making it extremely simple to implement the procedure has shown a considerable reduction in the number of iterations for grids with large graph depth the savings exceed an order of magnitude furthermore the technique has shown a remarkable insensitivity to the number of groups regions chosen and to the way the groups are formed

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

a discrete vector potential model for unsteady incompressible viscous flows abstract a recent approach to generate a zero divergence velocity field by operating directly on the discretized navier stokes equations is used to obtain the decoupling of the pressure from the velocity field by following the methodology suggested by amit hall and porsching the feasibility of treating three dimensional flows and multiply connected domains is analyzed the present model keeps the main features of the classical vector potential method in that it generates a divergence free velocity field through an algebraic manipulation of the discrete equations at the same time the boundary conditions are still imposed on the discrete values of the primitive variables the accuracy of the method is tested against the exact solution for a recirculating unsteady flow both in simply and doubly connected domains several applications to flow fields in three dimensional enclosures or in multiply connected domains are presented and discussed in terms of accuracy and efficiency of the method

from the trapezoidal rule to higher order accurate and unconditionally stable time integration method for structural dynamics abstract since the trapezoidal rule was originally proposed as the constant average acceleration method by newmark it has been modified to various forms however most of the modifications were limited to its numerical coefficients and as a result little change in the characteristics was obtained like numerical dissipation with the loss of accuracy in this work the constant average acceleration which is a fundamental feature of the trapezoidal rule is investigated and generalized using the generalization of average acceleration concept a higher order accurate and unconditionally stable time integration method is developed the stability analysis is carried out for a system of single degree of freedom the analysis proves that the present method is unconditionally stable to observe the accuracy of the method several numerical tests are performed and the results are compared with the results from the trapezoidal rule and the exact solution from the numerical tests it has been known that the present method has a higher order accuracy

a methodology for the generation of low cost higher order methods for linear dynamics this work presents a methodology which generates efficient higher order methods for linear dynamics by improving the accuracy properties of norsett methods towards those of pade methods the methodology is based on a simple and low cost iterative procedure which is used to implement a set of higher order methods with controllable dissipation a sequence of improved solutions is obtained which correspond to algorithms offering an effective compromise between the efficiency of norsett methods and the accuracy of pade methods moreover a direct control over high frequency dissipation is possible by means of an algorithmic parameter numerical tests are reported which confirm that this set of algorithms is really attractive for linear dynamic analysis copyright 2003 john wiley sons ltd

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

a monotonicity preserving conservative sharp interface flow solver for high density ratio two phase flows this paper presents a novel approach for solving the conservative form of the incompressible two phase navier stokes equations in order to overcome the numerical instability induced by the potentially large density ratio encountered across the interface the proposed method includes a volume of fluid type integration of the convective momentum transport a monotonicity preserving momentum rescaling and a consistent and conservative ghost fluid projection that includes surface tension effects the numerical dissipation inherent in the volume of fluid treatment of the convective transport is localized in the interface vicinity enabling the use of a kinetic energy conserving discretization away from the singularity two and three dimensional tests are presented and the solutions shown to remain accurate at arbitrary density ratios the proposed method is then successfully used to perform the detailed simulation of a round water jet emerging in quiescent air therefore suggesting the applicability of the proposed algorithm to the computation of realistic turbulent atomization

a weighted meshfree collocation method for incompressible flows using radial basis functions abstract a weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible navier stokes equations the velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability no artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation radial basis collocation method in a lagrangian form is quite easy to capture the moving boundary or free surface in flow problems moreover solid boundary conditions can be enforced directly and no special treatments are required further critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated numerical studies validate the high accuracy as well as good stability of the presented method

solution of linear elastodynamics problems with space time finite elements on structured and unstructured meshes abstract a new implicit approach for the solution of linear elastodynamics problems on structured and unstructured meshes with space time finite elements is suggested new weak formulations are based on time continuous tcg and discontinuous tdg galerkin methods it was shown that for structured meshes the new tcg method has a higher order of accuracy by a factor of two than that of the standard tcg method at the same number of degrees of freedom the accuracy of the new tcg method is one order higher than the accuracy of the standard tdg method the introduction of new weighting functions for the weak formulations of the tcg and tdg methods allows control of the high frequency behavior without degradation of accuracy of the methods a variable number of elements in the time direction as well as a variable order of polynomial time approximations of finite elements for time slabs allows the implementation of a coupled space time adaptive procedure using structured or unstructured meshes numerical examples for 1 d problems demonstrate the effectiveness of the technique

finite element simulation of wave propagation in an axisymmetric bar an accurate solution for high frequency pulse propagation in an axisymmetric elastic bar is obtained using a new finite element technique that yields accurate non oscillatory solutions for wave propagation problems in solids the solution of the problem is very important for the understanding of dynamics experiments in the split hopkinson pressure bar shpb in contrast to known approaches no additional assumptions are necessary for the accurate solution of the considered problem the new solution helps to elucidate the complicated distribution of parameters during high frequency pulse propagation down the bar as well as to estimate the applicability of the traditional dispersion correction used in the literature for the analysis of wave propagation in a finite bar due to the dimensionless formulation of the problem the numerical results obtained depend on poisson s ratio the length of the bar and the pulse frequency and are independent of young s modulus the density and the radius of the bar

a fast pressure correction method for incompressible two fluid flows we have developed a new pressure correction method for simulating incompressible two fluid flows with large density and viscosity ratios the method s main advantage is that the variable coefficient poisson equation that arises in solving the incompressible navier stokes equations for two fluid flows is reduced to a constant coefficient equation which can be solved with an fft based fast poisson solver this reduction is achieved by splitting the variable density pressure gradient term in the governing equations the validity of this splitting is demonstrated from our numerical tests and it is explained from a physical viewpoint in this paper the new pressure correction method is coupled with a mass conserving volume of fluid method to capture the motion of the interface between the two fluids but in general it could be coupled with other interface advection methods such as level set phase field or front tracking first we verified the new pressure correction method using the capillary wave test case up to density and viscosity ratios of 10 000 then we validated the method by simulating the motion of a falling water droplet in air and comparing the droplet terminal velocity with an experimental value next the method is shown to be second order accurate in space and time independent of the vof method and it conserves mass momentum and kinetic energy in the inviscid limit also we show that for solving the two fluid navier stokes equations the method is 10 40 times faster than the standard pressure correction method which uses multigrid to solve the variable coefficient poisson equation finally we show that the method is capable of performing fully resolved direct numerical simulation dns of droplet laden isotropic turbulence with thousands of droplets using a computational mesh of 1024 3 points

b spline method and zonal grids for simulations of complex turbulent flows a numerical technique for computations of turbulent flows is described the technique is based on b splines and allows grid embedding in physically significant flow regions numerical tests which include solutions of nonlinear advection diffusion equations and computations of flow over a circular cylinder at reynolds numbers up to 300 indicate that the method is accurate and efficient in computations of flow over a cylinder the lift drag and base suction coefficients agree well with existing experimental data and previous numerical simulations

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

prediction of wall pressure fluctuation in turbulent flows with an immersed boundary method the objective of this paper is to assess the accuracy and efficiency of the immersed boundary ib method to predict the wall pressure fluctuations in turbulent flows where the flow dynamics in the near wall region is fundamental to correctly predict the overall flow the present approach achieves sufficient accuracy at the immersed boundary and overcomes deficiencies in previous ib methods by introducing additional constraints a compatibility for the interpolated velocity boundary condition related to mass conservation and the formal decoupling of the pressure on this surfaces the immersed boundary approximated domain method ib adm developed in the present study satisfies these conditions with an inexpensive computational overhead the ib adm correctly predicts the near wall velocity pressure and scalar fields in several example problems including flows around a very thin solid object for which incorrect results were obtained with previous ib methods in order to have sufficient near wall mesh resolution for les and dns computations the present approach uses local mesh refinement the present method has been also successfully applied to computation of the wall pressure space time correlation in dns of turbulent channel flow on grids not aligned with the boundaries when applied to a turbulent flow around an airfoil the computed flow statistics the mean rms flow field and power spectra of the wall pressure are in good agreement with experiment

an implicit forcing immersed boundary method for simulating viscous flows in irregular domains abstract we present a method for solving the incompressible navier stokes equations in irregular domains these equations are discretized using finite difference method in a uniform cartesian grid stationary rigid boundaries are embedded in the cartesian grid and singular forces are applied at the rigid boundaries to impose the no slip conditions the singular forces are then distributed to the nearby cartesian grid points using the immersed boundary method in the present work the singular forces are computed implicitly by solving a small system of equations at each time step this system of equations is derived from a second order projection method the main advantage of this method is that it imposes the no slip boundary condition exactly and avoids the need for small time step to maintain stability the ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2 dimensional flows past a circular cylinder multiple rigid obstacles and 3 dimensional flow past a sphere

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

implementation and adaptivity of a space time finite element method for structural dynamics abstract in this paper we study the implementation and adaptivity of a space time finite element method for the analysis of two dimensional problems in structural dynamics the method also known as the time discontinuous galerkin method or the dg method employs finite elements discretizations in space and time simultaneously with bilinear basis functions that are continuous in space and discontinuous in time based on using the zienkiewicz zhu error estimate in space and the jumps discontinuities of displacements and velocities in the total energy norm as a local error estimate in time an h adaptive procedure which updates the spatial mesh and the time step size automatically so as to control the estimated errors within specified tolerances is proposed numerical examples are presented showing that the considered dg method is of second order accuracy in space in l 2 and third order accuracy in time and the adaptive procedure is capable of updating the spatial meshes and the time steps when necessary making the solutions reliable and the computation efficient

structural dynamic analysis by a time discontinuous galerkin finite element method this paper studies a time discontinuous galerkin finite element method for structural dynamic problems by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels a new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations by using the jumps of the displacements and the velocities in the total energy norm as error indicators an adaptive time stepping procedure for selecting the proper time step size is described numerical examples including both single dof and multi dof problems are used to illustrate the performance of these algorithms comparisons with the exact results and or the results by the newmark integration scheme are given it is shown that the time discontinuous galerkin finite element method discussed in this study possesses good accuracy third order and stability properties its numerical implementation is not difficult and the higher computational cost needed in each time step is compensated by use of a larger time step size

numerical simulation of isotropic turbulence using a collocated approach and a nonorthogonal grid system we studied the effect of using a nonorthogonal grid coordinate system and a finite volume approach in the simulation of decaying isotropic turbulence calculations were performed in distorted periodic cubic boxes and with a turbulent reynolds number based on the taylor microscale and on a root mean square turbulent velocity of approximately 40 a preliminary study showed that in the nonorthogonal grids some fourier modes of the discretized derivatives can have greater amplitude or the phase inverted relatively to the modes of the exact derivative contrary to what occurs with a cartesian grid system however in the simulations the statistical distributions of velocity pressure and longitudinal and lateral velocity derivatives were always identical regardless of the grid distortion the temporal evolution of the energy was also similar and the differences at the end of the simulations after about two eddy turnover times did not exceed 1 furthermore the grid nonorthogonality affected neither the isotropy of the fields nor the correlation between the vorticity and the principal rates of strain we concluded that the finite volume approach in nonorthogonal grid systems may be used in the numerical simulation of complex turbulent flows with either the direct numerical simulation or large eddy simulation methodologies

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

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

data aided and non data aided maximum likelihood snr estimators for cpm data aided da and non data aided nda maximum likelihood ml estimators for the snr of cpm in the presence of phase and frequency offset are derived and analyzed cramer rao bounds for both are obtained and compared to the simulated performance of these estimators for a full response example cpsfk and a partial response example gmsk analysis and simulations show that the performance of the da ml estimator suffers from an unremovable bias caused by uncompensated frequency offset due to frequency offset estimation errors as a consequence the estimator error variance of the da ml estimator is not able to achieve its lower bound in contrast the estimator error variance of the nda ml estimator is capable of achieving its lower bound although the lower bound for nda ml estimator is higher than the lower bound for the da ml estimator this is because the nda ml estimator is not burdened with the requirement of relying on estimates of nuisance parameters the nda estimator only achieves its lower bound when the observation length or true snr are sufficiently large

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

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

a note on the use of central schemes for incompressible navier stokes flows recently central methods combined with eno limiting 1 3 have become very popular for hyperbolic problems the main advantage is the simplicity of this method since no riemann problem has to be solved the only information necessary to know from the system under consideration is an estimate of the spectral radius of the linearization of the flux corresponding to the maximum wave speeds of the underlying system therefore the method is attractive also for problems where the riemann or approximate riemann problem is too difficult to solve or to implement this method has also been applied to the incompressible navier stokes equation in two dimensions using the vorticity stream function approach 3 4 this approach has been questioned by nielsen and naulin 5 they compared the cweno scheme as introduced by kurganov and levy section 5 example 5 3 with a standard spectral scheme and a finite difference approach using the arakawa 6 discretization the comparison focused on the conservation of integral quantities as the total energy and the total enstrophy their conclusion was that the spectral and arakawa scheme outperformed the cweno scheme quite dramatically in respect of numerical dissipation the result is that both the arakawa and the spectral scheme converge to the true solution from above in respect to the global quantities energy and enstrophy are too high if underresolved whereas the cweno scheme converges more slowly from below energy and enstrophy are too low nielsen and naulin 5 did not compare the amount of spurious oscillations where the cweno proves to have much better properties here we demonstrate that simply switching from the stream function approach to the integration of the primitive variables u with a projection method as discussed by brown et al 9 reduces the numerical dissipation quite substantially so that the cweno scheme approaches the properties of the spectral and the arakawa scheme without producing oscillations near strong vortex sheets for a second order scheme

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

a stabilized nitsche cut finite element method for the oseen problem abstract we provide the numerical analysis for a nitsche based cut finite element formulation for the oseen problem which has been originally presented for the incompressible navier stokes equations by schott and wall 2014 and allows the boundary of the domain to cut through the elements of an easy to generate background mesh the formulation is based on the continuous interior penalty cip method of burman et al 2006 which penalizes jumps of velocity and pressure gradients over inter element faces to counteract instabilities arising for high local reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure since the mesh does not fit the boundary dirichlet boundary conditions are imposed weakly by a stabilized nitsche type approach the addition of cip like ghost penalties in the boundary zone allows to prove that our method is inf supstable and to derive optimal order a priori error estimates in an energy type norm irrespective of how the boundary cuts the underlying mesh all applied stabilization techniques are developed with particular emphasis on low and high reynolds numbers two and three dimensional numerical examples corroborate the theoretical findings finally the proposed method is applied to solve the transient incompressible navier stokes equations on a complex geometry

semi implicit and explicit finite element schemes for coupled fluid thermal problems a comparative investigation based on a series of numerical tests of two purely explicit and one semi implicit finite element methods used for incompressible flow computation is presented the segregated approach is followed and the equations of motion are considered sequentially the fundamental concepts and characteristics of the formulations and the solution methodology used are described in technical detail various modifications to chorin s projection algorithm are investigated particularly with respect to their effects on stability and accuracy the stability of the semi implicit method is shown to be less restrictive when compared to the explicit methods as the reynolds number increases at large time steps the artificial viscosity is also reduced and higher accuracy is obtained the performance of the methods discussed in this paper is illustrated by the numerical solutions obtained for the cavity flow and flow past a rearward facing step problems at high reynolds numbers and free convection flow problem at high rayleigh numbers it is shown that the semi implicit method needs fewer iterations than the explicit methods and the accuracy of the present methods is guaranteed by comparison with the existing methods

linear stability of incompressible flow using a mixed finite element method we analyzed the linear stability for incompressible flow using the mixed finite element method first by means of the fractional step finite element method the two dimensional base flows were numerically computed over a range of reynolds numbers and then they were perturbed with three dimensional disturbances by using linear stability and normal mode analysis we obtained the partial differential equations governing the evolution of perturbation from linearized navier stokes equations with slight compressibility in terms of the mixed finite element discretization in which six node quadratic lagrange triangular elements with quadratic interpolation for velocities p2 and three node linear lagrange triangular elements for pressure p1 were used a nonsingular generalized eigenproblem was formulated from these equations whose solution gives the dispersion relationship between the complex growth rate and the wave number we examined the stability of lid driven cavity flow in this study the stability results obtained from thep2p1elements were compared with those fromp1p1elements the spatial distribution of the pressure disturbance can be smoothed by using ap2p1element such as that in the simulation of incompressible flows by the mixed finite element method however the choice of elemental type has little influence on the stability results this study presents stability curves to identify the critical reynolds number and the critical wavelength of the neutral mode and discusses the instability mechanism the taylor goertler like vortices in the cavity were obtained by means of the reconstruction of three dimensional flows the stability results and the reconstructed three dimensional flows were in agreement with the observed ones and the other numerical stability results respectively

time dependent viscous incompressible navier stokes equations the finite difference galerkin formulation and streamfunction algorithms abstract the finite difference galerkin fdg method is extended to time dependent incompressible navier stokes equations two algorithm development examples are given that use a staggered grid and centered differencing scheme for the primitive variables mass balance is used to solve the essential problems associated with applying the fdg method the use of the fdg method with this underlying discretization is shows to be the discrete analog of the continuum manipulations that lead to the fourth order streamfunction equation asymptotic and time evolution results obtained with a crank nicolson adams basforth algorithm are compared with published computations for re 400 1000 and 3200

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

explicit time domain approaches based on numerical green s functions computed by finite differences the exga family the present paper describes a new family of time stepping methods to integrate dynamic equations of motion the scalar wave equation is considered here however the method can be applied to time domain analyses of other hyperbolic e g elastodynamics or parabolic e g transient diffusion problems the algorithms presented require the knowledge of the green s function of mechanical systems in nodal coordinates the finite difference method is used here to compute numerically the problem green s function however any other numerical method can be employed e g finite elements finite volumes etc the green s matrix and its time derivative are computed explicitly through the range 0 dt with either the fourth order runge kutta algorithm or the central difference scheme in order to improve the stability of the algorithm based on central differences an additional matrix called step response is also calculated the new methods become more stable and accurate when a sub stepping procedure is adopted to obtain the green s and step response matrices and their time derivatives at the end of the time step three numerical examples are presented to illustrate the high precision of the present approach

projection methods for two velocity two pressure models for flows of heterogeneous mixtures in this paper a numerical analysis of two velocity two pressure models for flows of solid particles and fluids is presented first a formal exploitation of the weak formulation of such models asserts that they are amenable to integration via projection methods the challenging issues in the algorithm development for these models are then documented and suitable numerical methodologies for their remedy are devised subsequently an algorithm for the integration of the models of interest is proposed this is a two phase projection method on collocated grids that utilizes a fractional step time marching scheme it is further endowed with an interface detection and treatment methodology to properly account for the stiffness induced by the presence of moving and deforming material interfaces the efficiency and robustness of the proposed numerical method are assessed in a series of numerical experiments that are delineated in the last part of this paper

a fixed grid sharp interface method for bubble dynamics and phase change abstract a numerical method has been developed for direct simulation of bubble dynamics with large liquid to vapor density ratio and phase change the numerical techniques are based on a fixed grid finite volume method capable of treating the interface as a sharp discontinuity the unsteady axisymmetric navier stokes equations and energy equation in both liquid and vapor phases are computed the mass momentum and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement the cubic b spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures nondimensional parameters including reynolds weber and jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics based on the present sharp interface approach bubble dynamics for density ratio of 1600 or higher with and without phase change can be successfully computed