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

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

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

a method to discern complexity in two dimensional patterns generated by coupled map lattices complex patterns generated by the time evolution of a one dimensional digitalized coupled map lattice are quantitatively analyzed a method for discerning complexity among different patterns is implemented the quantitative results indicate two zones in parameter space where the dynamics show the most complex patterns these zones are located on the two edges of an absorbent region where the system displays spatio temporal intermittency

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

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

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

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

periodic and localized states in natural doubly diffusive convection numerical continuation is used to follow branches of steady doubly diffusive convection in a vertical slot driven by imposed horizontal temperature and concentration gradients no slip boundary conditions are used on the lateral walls periodic boundary conditions with large spatial period are used in the vertical direction a variety of different states both spatially periodic and spatially localized are identified and the profusion of the resulting solution branches is linked to a phenomenon known as homoclinic snaking

non systematic complex number rs coded ofdm by unique word prefix in this paper we expand our recently introduced concept of unique word orthogonal frequency division multiplexing uw ofdm in uw ofdm the cyclic prefixes cps are replaced by deterministic sequences the so called unique words uws the uws are generated by appropriately loading a set of redundant subcarriers by that a systematic complex number reed solomon rs code construction is introduced in a quite natural way because an rs code may be defined as the set of vectors for which a block of successive zeros occurs in the other domain w r t a discrete fourier transform for a fixed block different to zero i e a uw a coset code of an rs code is generated a remaining problem in the original systematic coded uw ofdm concept is the fact that the redundant subcarrier symbols disproportionately contribute to the mean ofdm symbol energy in this paper we introduce the concept of non systematic coded uw ofdm where the redundancy is no longer allocated to dedicated subcarriers but distributed over all subcarriers we derive optimum complex valued code generator matrices matched to the best linear unbiased estimator blue and to the linear minimum mean square error lmmse data estimator respectively with the help of simulations we highlight the advantageous spectral properties and the superior bit error ratio ber performance of non systematic coded uw ofdm compared to systematic coded uw ofdm and to cp ofdm in additive white gaussian noise awgn as well as in frequency selective environments

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

a fisher gradient complexity in systems with spatio temporal dynamics abstract we define a benchmark for definitions of complexity in systems with spatio temporal dynamics and employ it in the study of collective motion we show that lmc s complexity displays interesting properties in such systems while a statistical complexity model scm based on autocorrelation reasonably meets our perception of complexity however this scm is not as general as desirable as it does not merely depend on the system s probability distribution function inspired by the notion of fisher information we develop a scm candidate which we call the fisher gradient complexity which exhibits nice properties from the viewpoint of our benchmark

complex field coding for ofdm over fading wireless channels orthogonal frequency division multiplexing ofdm converts a time dispersive channel into parallel subchannels and thus facilitates equalization and de coding but when the channel has nulls close to or on the fast fourier transform fft grid uncoded ofdm faces serious symbol recovery problems as an alternative to various error control coding techniques that have been proposed to ameliorate the problem we perform complex field coding cfc before the symbols are multiplexed we quantify the maximum achievable diversity order for independent and identically distributed i i d or correlated rayleigh fading channels and also provide design rules for achieving the maximum diversity order the maximum coding gain is given and the encoder enabling the maximum coding gain is also found simulated performance comparisons of cfc ofdm with existing block and convolutionally coded ofdm alternatives favor cfc ofdm for the code rates used in a hiperlan2 experiment

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

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

a direct method for computation of simple bifurcations in the present study we focus on the computational analysis of partial differential equations with emphasis on the stability of the equilibrium states and on their bifurcations in practical applications it is not sufficient to obtain an equilibrium solution at a point in the parameter space the equilibrium solution branches their stability characteristics and particularly the critical points of transition from one state to another e g bifurcation points are required for understanding the physics of the problem in principle the linear stability of an equilibrium state can be investigated by solving an eigenvalue problem and consequently the points of bifurcations can be detected we review alternative techniques for detecting bifurcation points which are direct and numerically efficient and therefore more practical starting with a large dimension dynamical system which represents a projection of a set of coupled partial differential equations onto a basis function we discuss the relative effectiveness of the time evolution approach the test function approach and the direct method we will then extend the direct method for a more practical and efficient implementation with this technique we compute the sequence of transitions from steady state to chaotic flow in a two dimensional lid driven cavity of aspect ratio 0 8 1 0 and 1 5 we demonstrate the effectiveness of this technique by computing interesting new dynamics in this relatively simple hydrodynamic system in particular we show that depending on the aspect ratio the first transition from steady state could be through a supercritical or a subcritical hopf bifurcation leading the system to a time periodic state we construct the destabilizing disturbance structure and conclude that the first bifurcation of the primary steady state is due to the centrifugal instability of the primary eddy the mechanism of transition to chaos is low dimensional the transition to chaos occurs after a secondary hopf bifurcation

the use of symmetry in bifurcation calculations and its application to the be nard problem abstract some aspects of the use of symmetry in bifurcation calculations are discussed first it is shown how substantial reductions in cost are obtained in computing symmetry breaking bifurcation points by exploiting the underlying symmetry of a problem this is demonstrated in a finite element calculation of 2 dimensional benard convection in a finite cavity second the stability of paths of symmetry breaking bifurcation points which occur when a second parameter in the problem varies is investigated a criterion is established for deciding whether intersection of such paths is allowed by symmetry constraints

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

classical and bayesian linear data estimators for unique word ofdm unique word orthogonal frequency division multiplexing uw ofdm is a novel ofdm signaling concept where the guard interval is built of a deterministic sequence the so called unique word instead of the conventional random cyclic prefix in contrast to previous attempts with deterministic sequences in the guard interval the addressed uw ofdm signaling approach introduces correlations between the subcarrier symbols which can be exploited by the receiver in order to improve the bit error ratio performance in this paper we develop several linear data estimators specifically designed for uw ofdm some based on classical and some based on bayesian estimation theory furthermore we derive complexity optimized versions of these estimators and we study their individual complex multiplication count in detail finally we evaluate the estimators performance for the additive white gaussian noise channel as well as for selected indoor multipath channel scenarios

theoretical performance evaluation and optimization of uw ofdm unique word uw ofdm is a new ofdm variant that offers several advantages over cyclic prefix ofdm such as better bit error rate ber performance in frequency selective channels and much lower out of band radiation however all results with respect to the ber performance are based on monte carlo simulations only and offer little insight in the behaviour of the system performance as function of its parameters in this paper we theoretically analyse the performance of the uw ofdm system for both a deterministic channel and a fading channel and offer a systematic way to generate the uw ofdm signal to optimize the performance of the system when the fading channel is known at the transmitter we show that we can obtain a diversity order that is of the same order of magnitude as the theoretically maximum diversity order but when the channel is not known at the transmitter it will be very hard to achieve maximum diversity order although the obtained diversity order is larger than one

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

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

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

transitions between symmetric and nonsymmetric regimes in binary mixture convection abstract we present here a comprehensive picture of the different bifurcations found for small to moderate rayleigh number in binary mixture convection with lateral heating and negative separation ratio s the present work connects the symmetric regime found for pure fluid s 0 mercader et al 2005 with the fundamentally nonsymmetric regime found for s 1 meca et al 2004 2 3 we give a global context as well as an interpretation for the different associations of bifurcations found and in particular we interpret an association of codimension two bifurcations in terms of a higher codimension bifurcation never found to our knowledge in the study of an extended system

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

the computation of symmetry breaking bifurcation points we consider symmetry breaking simple singular points of a parameter dependent nonlinear equation g lambda x 0 such points are intimately connected to pitchfork bifurcation points and we sh

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

marangoni benard convection with a deformable free surface computations of marangoni convection are usually performed in two or three dimensional domains with rigid boundaries in two dimensions allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure we describe a finite element technique for calculating bifurcations that arise due to thermal gradients in a two dimensional domain with a deformable free surface the fluid is assumed to be newtonian to conform to the boussinesq approximation and to have a surface tension that varies linearly with temperature an orthogonal mapping from the physical domain to a reference domain is employed which is determined as the solution to a pair of elliptic partial differential equations the mapping equations and the equilibrium equations for the velocity pressure and temperature fields and their appropriate nonlinear boundary conditions are discretized using the finite element method and solved simultaneously by newton iteration contact angles other than 90 degrees are shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner the loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to and less than 90 degrees

chernoff type bounds for the gaussian error function we study single term exponential type bounds also known as chernoff type bounds on the gaussian error function this type of bound is analytically the simplest such that the performance metrics in most fading channel models can be expressed in a concise closed form we derive the conditions for a general single term exponential function to be an upper or lower bound on the gaussian error function we prove that there exists no tighter single term exponential upper bound beyond the chernoff bound employing a factor of one half regarding the lower bound we prove that the single term exponential lower bound of this letter outperforms previous work numerical results show that the tightness of our lower bound is comparable to that of previous work employing eight exponential terms

nonlinear doubly diffusive convection in vertical enclosures nonlinear doubly diffusive convection in two dimensional enclosures driven by lateral temperature and concentration differences is studied using a combination of analytical and numerical techniques the study is organized around a special case that allows a static equilibrium the stationary states that bifurcate from this equilibrium are either symmetric or antisymmetric with respect to diagonal reflection local bifurcation analysis around the critical aspect ratio at which both modes appear simultaneously is complemented using numerical continuation perturbation of this situation to one in which no static equilibrium exists provides important information about the multiplicity of steady states in this system

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

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

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

different modes of rayleigh benard instability in two and three dimensional rectangular enclosures the article describes a complete numerical solution of a recently formulated benchmark problem devoted to the parametric study of rayleigh benard instability in rectangular two and three dimensional boxes the solution is carried out by the spectral galerkin method with globally defined three dimensional divergent free basis functions which satisfy all boundary conditions the general description of these three dimensional basis functions which can be used for a rather wide spectrum of problems is presented the results of the parametric calculations are presented as neutral curves showing the dependence of the critical rayleigh number on the aspect ratio of the cavity the neutral curves consist of several continuous branches which belong to different modes of the most dangerous perturbation the patterns of different perturbations are also reported the results obtained lead to some new conclusions about the patterns of the most dangerous perturbations and about the similarities between two and three dimensional models some extensions of the considered benchmark problem are discussed

the onset of convection in a finite container due to surface tension and buoyancy abstract a method for predicting instabilities which combines recent techniques from bifurcation theory with the finite element method is described it is applied to the prediction of the onset of convection driven by both surface tension and buoyancy in rectangular containers for zero buoyancy the critical values of the marangoni number for the first two bifurcations from the trivial solution are found for a two dimensional cavity of aspect ratio 2 the variation of these critical values with aspect ratio is obtained by continuation methods and this reveals an interlacing of modes as the container size increases it is established that the bifurcation to an even number of cells is transcritical rather than pitchfork and the turning point on the subcritical branch is located as a function of aspect ratio the hysteresis associated with the transcritical bifurcation is small as the surface tension forces decrease to zero so that the convection is driven by buoyancy alone the amount of hysteresis associated with the transcritical bifurcation becomes vanishingly small the reason for this is not fully understood

statistical complexity measures as telltale of relevant scales in emergent dynamics of spatial systems abstract the definition of complexity through statistical complexity measures scm has recently seen major improvements mostly the effort is concentrated in measures on time series we propose a scm definition for spatial dynamical systems our definition is in line with the trend to combine entropy with measures of structure such as disequilibrium we study the behaviour of our definition against the vectorial noise model of collective motion from a global perspective we show how our scm is minimal at both the microscale and macroscale while it reaches a maximum at the ranges that define the mesoscale in this model from a local perspective the scm is minimum both in highly ordered and disordered areas while it reaches a maximum at the edges between such areas these characteristics suggest this is a good candidate for detecting the mesoscale of arbitrary dynamical systems as well as regions where the complexity is maximal in such systems