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

telescopic projective methods for parabolic differential equations projective methods were introduced in an earlier paper c w gear i g kevrekidis projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum nec research institute report 2001 029 available from http www neci nj nec com homepages cwg projective pdf abbreviated version to appear in sisc as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active in this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi discretization of pdes with parabolic components

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

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

algorithmically insoluble problems about finitely presented solvable groups lie and associative algebras 1 1 in 1955 novikov 111 constructed a finitely presented group with an insoluble word problem cf miller lo and also boone 8 shortly afterwards first adyan l 4 and then rabin 13 143 used this group of novikov to prove that for every one of a host of group theoretic properties there is no algorithm whereby one can decide whether any finitely presented group has such a property in particular there is no algorithm whereby one can decide whether any finitely presented group is of order 1 thus the isomorphism problem for finitely presented groups is algorithmically insoluble these negative results were followed in 1959 by a number of similar ones concerned with elements and subgroups of finitely presented groups baumslag boone and neumann s thus for example there is no algorithm whereby one can decide whether or not any element in a suitably chosen finitely presented group is a proper power in 1981 harlampovich 9 constructed a finitely presented solvable group of derived length 3 with an insoluble word problem this then raised the possibility that many of the results cited above for finitely presented groups can be carried over to finitely presented solvable groups it is not possible to carry over these results en masse to finitely presented solvable groups since it is easy to decide whether any finitely presented solvable group is of order 1 indeed there is an algorithm whereby one can determine whether any finitely presented solvable group is polycyclic baumslag cannonito and miller 6 again no such algorithm exists for finitely presented groups as a whole adyan 2 the methods employed previously cannot be applied to solvable groups because the constructions

computational study of subcritical response in flow past a circular cylinder flow past a circular cylinder is investigated in the subcritical regime below the onset of benard von karman vortex shedding at reynolds number re c similar or equal to 47 the transient response of infinitesimal perturbations is computed the domain requirements for obtaining converged results is discussed at length it is shown that energy amplification occurs as low as re 2 2 throughout much of the subcritical regime the maximum energy amplification increases approximately exponentially in the square of re reaching 6800 at re c the spatiotemporal structure of the optimal transient dynamics is shown to be transitory benard von karman vortex streets at re similar or equal to 42 the long time structure switches from exponentially increasing downstream to exponentially decaying downstream three dimensional computations show that two dimensional structures dominate the energy growth except at short times

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

spectral distributed lagrange multiplier method algorithm and benchmark tests we extend the formulation of the distributed lagrange multiplier dlm approach for particulate flows to high order methods within the spectral hp element framework we implement the rigid body motion constraint inside the particle via a penalty method the high order dlm method demonstrates spectral convergence rate i e discretization errors decrease exponentially as the order of spectral polynomials increases we provide detailed comparisons between the spectral dlm method direct numerical simulations and the force coupling method for a number of 2d and 3d benchmark flow problems we also validate the spectral dlm method with available experimental data for a transient problem the new dlm method can potentially be very effective in many moving body problems where a smaller number of grid points is required in comparison with low order methods

a rotational velocity correction projection method for unsteady incompressible magnetohydrodynamics equations abstract this paper provides a rotational velocity correction projection method for the unsteady incompressible magnetohydrodynamics mhd equations in this method the electrical field variable e was eliminated in this way the gauss s law can be conserved the velocity variable u and the pressure field p are solved by a rotational velocity correction projection method the optimal error estimates of the l 2 norm will be given finally we shall present some numerical results to confirm our analysis it shows that our method is effective and can conserve the gauss s law very well

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

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

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 splitting technique of higher order for the navier stokes equations this article presents a splitting technique for solving the time dependent incompressible navier stokes equations using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time the integration of adaptive methods in space and time in the splitting are discussed in this algorithm a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm results on the flow around a cylinder s and the driven cavity s problem are presented

a new domain decomposition method with overlapping patches for ultrascale simulations application to biological flows we address the failure in scalability of large scale parallel simulations that are based on semi implicit time stepping and hence on the solution of linear systems on thousands of processors we develop a general algorithmic framework based on domain decomposition that removes the scalability limitations and leads to optimal allocation of available computational resources it is a non intrusive approach as it does not require modification of existing codes specifically we present here a two stage domain decomposition method for the navier stokes equations that combines features of discontinuous and continuous galerkin formulations at the first stage the domain is subdivided into overlapping patches and within each patch a c 0 spectral element discretization second stage is employed solution within each patch is obtained separately by applying an efficient parallel solver proper inter patch boundary conditions are developed to provide solution continuity while a multilevel communicating interface mci is developed to provide efficient communication between the non overlapping groups of processors of each patch the overall strong scaling of the method depends on the number of patches and on the scalability of the standard solver within each patch this dual path to scalability provides great flexibility in balancing accuracy with parallel efficiency the accuracy of the method has been evaluated in solutions of steady and unsteady 3d flow problems including blood flow in the human intracranial arterial tree benchmarks on bluegene p cray xt5 and sun constellation linux cluster have demonstrated good performance on up to 96 000 cores solving up to 8 21b degrees of freedom in unsteady flow problem the proposed method is general and can be potentially used with other discretization methods or in other applications

x semipermutable subgroups of finite groups abstract let x be a non empty subset of a group g then we call a subgroup a of g a x semipermutable subgroup of g if a has a supplement t in g such that for every subgroup t 1 of t there exists an element x x such that a t 1 x t 1 x a in this paper we study the properties of x semipermutable subgroups in particular a new version of the famous schur zassenhaus theorem in terms of x semipermutable subgroups is given

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

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

a fourier legendre spectral element method in polar coordinates in this paper a new fourier legendre spectral element method based on the galerkin formulation is proposed to solve the poisson type equations in polar coordinates the 1 r singularity at r 0 is avoided by using gauss radau type quadrature points in order to break the time step restriction in the time dependent problems the clustering of collocation points near the pole is prevented through the technique of domain decomposition in the radial direction a number of poisson type equations subject to the dirichlet or neumann boundary condition are computed and compared with the results in literature which reveals a desirable result

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

a dffd simulation method combined with the spectral element method for solid fluid interaction problems a 2d simulation method for a rigid body moving in an incompressible viscous fluid is proposed it combines one of the immersed boundary methods the dffd direct forcing fictitious domain method with the spectral element method the former is employed for efficiently capturing the two way fsi fluid structure interaction and the geometric flexibility of the latter is utilized for any possibly co existing stationary and complicated solid or flow boundary a pseudo body force is imposed within the solid domain to enforce the rigid body motion and a lagrangian mesh composed of triangular elements is employed for tracing the rigid body in particular a so called sub cell scheme is proposed to smooth the discontinuity at the fluid solid interface and to execute integrations involving eulerian variables over the moving solid domain the accuracy of the proposed method is verified through an observed agreement of the simulation results of some typical flows with analytical solutions or existing literatures the direct forcing fictitious domain method and the spectral element method are combined direct interpolation of eulerian variables onto lagrangian mesh is not necessary non fluctuating pressure fields are obtained with a smoothed pseudo body force the proposed method possesses a spectral accuracy both small and large solid to fluid density ratios are applicable

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

analysis of artificial pressure equations in numerical simulations of a turbulent channel flow abstract recently several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation avoiding the resolution of a poisson equation these methods can be seen as various levels of approximation of the compressible navier stokes equation in the low mach number limit we study the simulation of incompressible wall bounded flows using several artificial pressure equations in order to determine the most relevant approximations the simulations are stable using a finite difference method in a staggered grid system even without diffusive term and converge to the incompressible solution both in direct numerical simulations and for coarser meshes to be used in large eddy simulations a pressure equation with a convective and a diffusive term produces a more accurate solution than a compressible solver or methods involving more approximations this suggests that it is near to an optimal level of approximation the presence of a convective term in the pressure evolution equation is in particular crucial for the accuracy of the method the rate of convergence of the solution in terms of artificial mach number is studied numerically and validates the theoretical quadratic convergence rate we demonstrate that this property can be used to accelerate the rate of convergence using an extrapolation in terms of artificial mach number since the approach is based on an explicit and local system of equations the numerical procedure is massively parallelisable and has low memory requirements

a weak legendre collocation spectral method for the solution of the incompressible navier stokes equations in unstructured quadrilateral subdomains a weak legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible navier stokes equations in general two dimensional and axisymmetric geometries a semi implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field a series of elliptic boundary value problems arises from the above procedure each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains in particular a modified variational formulation of the partial differential equations is presented which leads after discretization to a weak multidomain approximation of the corresponding problems a weak formalism for the influence matrix technique is also developed which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns a method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization exponential convergence is demonstrated for a plethora of stokes and navier stokes simulations

a high order discontinuous galerkin fourier incompressible 3d navier stokes solver with rotating sliding meshes we present the development of a sliding mesh capability for an unsteady high order order 3 h p discontinuous galerkin solver for the three dimensional incompressible navier stokes equations a high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh through the use of curved boundary elements and mixed triangular quadrilateral meshes a second order stiffly stable method is used to discretise in time the arbitrary lagrangian eulerian form of the incompressible navier stokes equations spatial discretisation is provided by the symmetric interior penalty galerkin formulation with modal basis functions in the x y plane allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques spatial discretisation in the z direction is provided by a purely spectral method that uses fourier series and allows computation of spanwise periodic three dimensional flows the developed solver is shown to provide high order solutions second order in time convergence rates and spectral convergence when solving the incompressible navier stokes equations on meshes where fixed and rotating elements coexist in addition an exact implementation of the no slip boundary condition is included for curved edges circular arcs and naca 4 digit airfoils where analytic expressions for the geometry are used to compute the required metrics the solver capabilities are tested for a number of two dimensional problems governed by the incompressible navier stokes equations on static and rotating meshes the taylor vortex problem a static and rotating symmetric naca0015 airfoil and flows through three bladed cross flow turbines in addition three dimensional flow solutions are demonstrated for a three bladed cross flow turbine and a circular cylinder shadowed by a pitching naca0012 airfoil

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

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

computable algebra and group embeddings this work grew out of our investigations of the infinitely generated subgroups of finitely presented groups a useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in l this technique was then applied in 1 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group it became apparent from 2 that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups in this paper these extensions are carried out with the consequent application that every countable locally abelian by nilpotent group can be embedded in a finitely presented group much of the effective commutative algebra we require seems to be known see for instance 6 9 14 171 unfortunately a systematic account is not available in particular we need a proof of the hilbert basis theorem which

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

long time stability of high order multistep numerical schemes for two dimensional incompressible navier stokes equations the long time stability properties of a few multistep numerical schemes for the two dimensional incompressible navier stokes equations formulated in vorticity stream function are investigated in this article these semi implicit numerical schemes use a combination of explicit adams bashforth extrapolation for the nonlinear convection term and implicit adams moulton interpolation for the viscous diffusion term up to fourth order accuracy in time as a result only two poisson solvers are needed at each time step to achieve the desired temporal accuracy the fully discrete schemes with fourier pseudospectral approximation in space are analyzed in detail with the help of a priori analysis and aliasing error control techniques we prove uniform in time bounds for these high order schemes in both l 2 and h m norms for m ge 1 provided that the time step is bounded by a given constant such a long time stability is also demonstrated by the numerical experiments

the algorithmic theory of polycyclic by finite groups a group is said to be polycyclic by finite or a pf group for short if it has a polycyclic normal subgroup of finite index equivalently pf groups are exactly the groups which have a series of finite length whose infinite factors are cyclic by a well known theorem of p hall every pf group is finitely presented and in fact pf groups form the largest known sectionclosed class of finitely presented groups it is this fact that makes pf groups natural objects of study from the algorithmic standpoint the general aim of the algorithmic theory of pf groups can be described as the collection of information about a pf group which can in principle

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

high order accurate solution of the incompressible navier stokes equations on massively parallel computers the emergence of petascale supercomputers requires us to develop today s simulation codes for incompressible flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power in that spirit we present a massively parallel high order navier stokes solver for large incompressible flow problems in three dimensions the governing equations are discretized with finite differences in space and a semi implicit time integration scheme this discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers the iterative solver for the pressure uses a highly efficient commutation based preconditioner which is robust with respect to grid stretching the efficiency of the implementation is further enhanced by carefully setting the adaptive termination criteria for the different iterative solvers the computational work is distributed to different processing units by a geometric data decomposition in all three dimensions this decomposition scheme ensures a low communication overhead and excellent scaling capabilities the discretization is thoroughly validated first we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow second we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria third orr sommerfeld and squire eigenmodes for plane poiseuille flow are simulated and compared to analytical results fourth the practical applicability of the implementation is tested for transitional and turbulent channel flow the results are compared to solutions from a pseudospectral solver subsequently the performance of the commutation based preconditioner for the pressure iteration is demonstrated finally the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to o 10 4 processing units and o 10 1 1 grid points

unsteady two dimensional flows in complex geometries comparative bifurcation studies with global eigenfunction expansions we present a bifurcation study of the incompressible navier stokes equations in a model complex geometry a spatially periodic array of cylinders in a channel the dynamics of the flow include a hopf bifurcation from steady to oscillatory flow at an approximate reynolds number r of 350 and the appearance of a second frequency at approximately r simeq 890 the multiple frequency dynamics include a substantial increase in spatial and temporal scales with reynolds number as compared with the simple limit cycle oscillation present close to r 350 numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions the first basis set is derived through principal factor analysis karhunen loeve expansion of snapshots from accurate direct spectral element numerical solutions of the navier stokes equations the second set is obtained from the eigenfunctions of the stokes operator for this geometry finally eigenfunctions are derived from a singular stokes operator i e the stokes operator modified to include a variable coefficient which vanishes at the domain boundaries truncated systems of sim 100 odes are obtained through projection of the navier stokes equations onto the basis sets and a comparative study of the resulting dynamical models is performed

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

on mass conservation and solvability of the discretized variable density zero mach navier stokes equations abstract this paper presents a study of a variable density zero mach navier stokes discretization and subtle difficulties associated with the pressure a consistent compatibility condition between the temperature and velocity is first derived it is shown that one can accurately and efficiently solve all primitive variables with the uniform order of accuracy of the specified discretization and time marching schemes it is shown that higher density ratios than those commonly reported are achievable and it is in part traced to the solvability of the pressure poisson equation numerical properties are assessed with the method of manufactured solutions and several classical testing cases are carried out to validate the code ranging from variable density mixing problems to premixed flames with forcing

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

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

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

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

kinetically reduced local navier stokes equations for simulation of incompressible viscous flows recently another approach to study incompressible fluid flow was suggested s ansumali i karlin and h ottinger phys rev lett 94 080602 2005 the kinetically reduced local navier stokes krlns equations we consider a simplified two dimensional krlns system and compare it with chorin s artificial compressibility method a comparison of the two methods for steady state computation of the flow in a lid driven cavity at various reynolds numbers shows that the results from both methods are in good agreement with each other however in the transient flow it is demonstrated that the krlns equations correctly describe the time evolution of the velocity and of the pressure unlike the artificial compressibility method

finite groups with given s embedded and n embedded subgroups abstract let g be a finite group and h a subgroup of g then h is said to be s permutable in g if h p p h for all sylow subgroups p of g let h s g be the subgroup of h generated by all those subgroups of h which are s permutable in g and h s g the intersection of all such s permutable subgroups of g which contain h we say that 1 h is s embedded in g if g has an s permutable subgroup t such that t h h s g and h t h s g 2 h is n embedded in g if g has a normal subgroup t such that t h h s g and h t h g our main results here are the following theorems theorem a a group g is supersoluble if and only if every maximal subgroup of every non cyclic sylow subgroup of the generalized fitting subgroup f g of g is n embedded in g theorem b a group g is supersoluble if and only if for every non cyclic sylow subgroup p of the generalized fitting subgroup f g of g every cyclic subgroup h of p with prime order and with order 4 if p is a non abelian 2 group and h z g is n embedded in g theorem f a group g is supersoluble if and only if every 2 maximal subgroup e of g with non primary index g e both has a cyclic supplement in e s g and is s embedded in g

spectral element formulations on non conforming grids a comparative study of pointwise matching and integral projection methods pointwise matching pm and integral projection ip methods are two widely used techniques to extend the classical weak formulations to include non conforming grids we present spectral element formulations on polynomial p type and geometric h type non conforming grids using both the pm also known as the constrained approximation and ip also known as the mortar element methods we systematically compare the convergence characteristics of pm and ip methods for diffusion convection and convection diffusion equations consistency errors due to the non conforming formulations of the diffusion equation result in convergence problems for the pm method using the maximum rule both non conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values causing convergence problems due to the consistency errors however small physical diffusion in the convection diffusion equation eliminates these problems resulting in spectral convergence for both methods encouraged by this we present spectral element formulations for incompressible navier stokes equations using pm and ip methods on p type and h type non conforming grids and demonstrate spectral convergence for unsteady and steady test cases results for two dimensional lid driven cavity flow at re 1000 are also presented

c normality of groups and its properties we introduce a new normality in a finite group called c normality we investigate the influence of c normal subgroups on the structure of finite groups in this paper we present some necessary and sufficient conditions for a group to be p nilpotent p supersolvable and supersolvable our results unify and generalize some related ones

subgroups permutable with all sylow subgroups fix a finite group g whenever s is a nonempty set of subgroups of g let sh denote the set of all subgroups h of g permutable with all s g s h so that the complex product hs s sh is their join it is obvious that s is closed with respect to joins in case s s syl g is the set of all sylow subgroups of g for all primes it is also closed with respect to intersec w x h tions kegel 4 so that syl g is a lattice the present paper may be seen as continuation of kegel s work to be more precise it aims to continue the first part of his work the second one being completed by w x kleidman 5 h syl g is the full subgroup lattice of g precisely when g g n is nilpotent the formation n of nilpotent groups will play a central role in general as usual we denote by g the nilpotent residual of g for subgroups h and k of g we write h s f h x and h k s k x g k 2 x x h x g k for the k core and the k closure of h respectively h s y1 x hx h k h suppose h g syl g then h and h do belong to syl g for all k n k but h usually does not nor does it the commutator group h9 nor the centre z h on the other hand we know that there is a unique subgroup of h minimal subject to containing h n and belonging to h syl g this is always a normal subgroup of g

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

the algorithmic theory of finitely generated metabelian groups algorithms are constructed which when an explicit presentation of a finitely generated metabelian group g in the variety x 2 is given produce finitary presentations for the derived subgroup g the centre z g the fitting subgroup fit g and the frattini subgroup 0 g additional algorithms of independent interest are developed for commutative algebra which construct the associated set of primes ass m of a finitely generated module m over a finitely generated commutative ring r and the intersection pr m of the maximal submodules of m

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 spectral deferred correction method for incompressible flow with variable viscosity abstract this paper presents a semi implicit spectral deferred correction sdc method for incompressible navier stokes problems with variable viscosity and time dependent boundary conditions the proposed method integrates elements of velocity and pressure correction schemes which yields a simpler pressure handling and a smaller splitting error than the sdpc method of minion and saye j comput phys 375 797 822 2018 combined with the discontinuous galerkin spectral element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space numerical experiments in three space dimensions demonstrate up to order 12 in time and 17 in space for constant as well as varying solution dependent viscosity compared to sdpc the present method yields a substantial improvement of accuracy and robustness against order reduction caused by time dependent boundary conditions

velocity correction schemes for the incompressible navier stokes equations in general coordinate systems this paper presents methods of including coordinate transformations into the solution of the incompressible navier stokes equations using the velocity correction scheme which is commonly used in the numerical solution of unsteady incompressible flows this is important when the transformation leads to symmetries that allow the use of more efficient numerical techniques like employing a fourier expansion to discretize a homogeneous direction two different approaches are presented in the first approach all the influence of the mapping is treated explicitly while in the second the mapping terms related to convection are treated explicitly with the pressure and viscous terms treated implicitly through numerical results we demonstrate how these methods maintain the accuracy of the underlying high order method and further apply the discretisation strategy to problems where mixed fourier spectral hp element discretisations can be applied thereby extending the usefulness of this discretisation technique

kinetically reduced local navier stokes equations an alternative approach to hydrodynamics an alternative approach the kinetically reduced local navier stokes krlns equations for the grand potential and the momentum is proposed for the simulation of low mach number flows the taylor green vortex flow is considered in the krlns framework and compared to the results of the direct numerical simulation of the incompressible navier stokes equations the excellent agreement between the krlns equations and the incompressible nonlocal navier stokes equations for this nontrivial time dependent flow indicates that the former is a viable alternative for computational fluid dynamics at low mach numbers

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

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

spectral elements on infinite domain a spectral element technique to approximate partial differential equations pdes on an infinite domain is examined the method is based on boyd s mapping of a semiinfinite interval to a finite interval and it is extended to a variational setting which allows for an implementation using a spectral element method by extending the method to a variational form a straightforward implementation allows for high order approximations over an infinite computational domain methods to find a high order approximation have been proposed by funaro maday mavriplis and many others the proposed methods all have drawbacks including restrictions on the behavior of the approximation at infinity because the proposed method is based on a mapping to a finite domain a much wider variety of boundary conditions can be enforced on the approximation

on weakly s permutable subgroups of finite groups abstract let g be a finite group h a subgroup of g and h s g the subgroup of h generated by all those subgroups of h which are s permutable in g then we say that h is weakly s permutable in g if g has a subnormal subgroup t such that h t g and t h h s g we fix in every non cyclic sylow subgroup p of g a subgroup d satisfying 1 d p and study the structure of g under the assumption that all subgroups h with h d are weakly s permutable in g

adaptive filtering for the lattice boltzmann method in this study a new selective filtering technique is proposed for the lattice boltzmann method this technique is based on an adaptive implementation of the selective filter coefficient the proposed model makes the latter coefficient dependent on the shear stress in order to restrict the use of the spatial filtering technique in sheared stress region where numerical instabilities may occur different parameters are tested on 2d test cases sensitive to numerical stability and on a 3d decaying taylorgreen vortex the results are compared to the classical static filtering technique and to the use of a standard subgrid scale model and give significant improvements in particular for low order filter consistent with the lbm stencil