the general properties of discrete time competitive dynamical systems abstract for strongly competitive discrete time dynamical systems on a strongly ordered topological vector space it is proved that any or limit set is unordered and lies on some invariant hypersurface with codimension one which generalizes hirsch s results for competitive autonomous systems of ordinary differential equations to competitive maps in a very general framework and implies the sarkovskii s theorem for planar strongly competitive and monotone maps we also show that the definition of competitiveness gives more restriction on homeomorphic property

an investigation into the characteristics of non linear frequency response functions part 1 understanding the higher dimensional frequency spaces the characteristics of generalized frequency response functions gfrfs of non linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques it is shown how a systematic analysis can be achieved for a wide class of non linear systems in the frequency domain using the proposed methods the paper is divided into two parts in part 1 the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi dimensional frequency spaces the visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time in part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non linear systems described by the narmax class of models case studies are included in part 2 to illustrate the application of the new methods

active control and drag optimization for flow past a circular cylinder the main objective of this article is to investigate computational methods for the active control and drag optimization of incompressible viscous flow past cylinders using the two dimensional navier stokes equations as the flow model the computational methodology relies on the following ingredients space discretization of the navier stokes equations by finite element approximations time discretization by a second order accurate two step implicit explicit finite difference scheme calculation of the cost function gradient by the adjoint equation approach and minimization of the cost function by a quasi newton method a la bfgs the above methods have been applied to boundary control by rotation of the flow around a circular cylinder and show 30 to 60 drag reduction compared to the fixed cylinder configuration for reynolds numbers in the range from 200 to 1000

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

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

accuracy analysis of explicit runge kutta methods applied to the incompressible navier stokes equations this paper investigates the temporal accuracy of the velocity and pressure when explicit runge kutta methods are applied to the incompressible navier stokes equations it is shown that at least up to and including fourth order the velocity attains the classical order of accuracy without further constraints however in case of a time dependent gradient operator which can appear in case of time varying meshes additional order conditions need to be satisfied to ensure the correct order of accuracy furthermore the pressure is only first order accurate unless additional order conditions are satisfied two new methods that lead to a second order accurate pressure are proposed which are applicable to a certain class of three and four stage methods a special case appears when the boundary conditions for the continuity equation are independent of time since in that case the pressure can be computed to the same accuracy as the velocity field without additional cost relevant computations of decaying vortices and of an actuator disk in a time dependent inflow support the analysis and the proposed methods

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

open and traction boundary conditions for the incompressible navier stokes equations we present numerical schemes for the incompressible navier stokes equations nse with open and traction boundary conditions we use pressure poisson equation ppe formulation and propose new boundary conditions for the pressure on the open or traction boundaries after replacing the divergence free constraint by this pressure poisson equation we obtain an unconstrained nse for stokes equation with open boundary condition on a simple domain we prove unconditional stability of a first order semi implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity using either boundary condition the schemes for the full nse that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and c 0 finite elements moreover when reynolds number is of o 1 and when the first order semi implicit time stepping is used time step size of o 1 is allowed in benchmark computations for the full nse besides standard stability and accuracy check various numerical results including flow over a backward facing step flow past a cylinder and flow in a bifurcated tube are reported numerically we have observed that using ppe formulation enables us to use the velocity pressure pairs that do not satisfy the standard inf sup compatibility condition our results extend that of johnston and liu h johnston j g liu accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term j comp phys 199 1 2004 221 259 which deals with no slip boundary conditions only

cech cohomology of attractors of discrete dynamical systems abstract let f r n r n be a homeomorphism and k an asymptotically stable attractor for f the aim of this paper is to study when the inclusion of k in its basin of attraction a k induces isomorphisms in cech cohomology we show that i this is true if coefficients are taken in q or z p p prime and ii it is true for integral cohomology if and only if the cech cohomology of k or a k is finitely generated we compute the cech cohomology of periodic point free attractors of volume contracting r 3 homeomorphisms and present applications to quite general models in population dynamics

energy preserving turbulent simulations at a reduced computational cost energy conserving discretizations are widely regarded as a fundamental requirement for high fidelity simulations of turbulent flows the skew symmetric splitting of the nonlinear term is a well known approach to obtain semi discrete conservation of energy in the inviscid limit however its computation is roughly twice as expensive as that of the divergence or advective forms alone a novel time advancement strategy that retains the conservation properties of skew symmetric based schemes at a reduced computational cost has been developed this method is based on properly constructed runge kutta schemes in which a different form advective or divergence for the convective term is adopted at each stage a general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew symmetric based techniques

modified immersed boundary method for flows over randomly rough surfaces abstract many phenomena ranging from biology to electronics involve flow over rough or irregular surfaces we treat such surfaces as random fields and use an immersed boundary method ibm with discrete random forcing to solve resulting stochastic flow problems our approach relies on the uhlmann formulation of the fluid solid interaction force computational savings stem from the modification of the time advancement scheme that obviates the need to solve the poisson equation for pressure at each sub step we start by testing the proposed algorithm on two classical benchmark problems the first deals with the wannier problem of stokesian flow around a cylinder in the vicinity of a moving plate the second problem considers steady state and transient flows over a stationary circular cylinder our simulation results show that our algorithm achieves second order temporal accuracy it is faster than the original ibm while yielding consistent estimates of such quantities of interest as the drag and lift coefficients the length of a recirculation zone in a cylinder s wake and the strouhal number then we use the proposed ibm algorithm to model flow over cylinders whose surface is either deterministically corrugated or randomly rough

an adaptive implicit explicit scheme for the dns and les of compressible flows on unstructured grids an adaptive implicit explicit scheme for direct numerical simulation dns and large eddy simulation les of compressible turbulent flows on unstructured grids is developed the method uses a node based finite volume discretization with summation by parts sbp property which in conjunction with simultaneous approximation terms sat for imposing boundary conditions leads to a linearly stable semi discrete scheme the solution is marched in time using an implicit explicit runge kutta imex rk time advancement scheme a novel adaptive algorithm for splitting the system into implicit and explicit sets is developed the method is validated using several canonical laminar and turbulent flows load balance for the new scheme is achieved by a dual constraint domain decomposition algorithm the scalability and computational efficiency of the method is investigated and memory savings compared with a fully implicit method is demonstrated a notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed

the design of energy transfer filters for energy focus filtering this paper is concerned with extending the energy transfer filters etfs designs to a more general case where the design requirement is to focus signal energy from different input frequency bands into a desired output frequency range etfs are a class of non linear filters recently proposed by the authors to employ non linear effects to transfer signal energy from one frequency band to a different frequency location the new etf designs considered in the present study involve the development of an algorithm for determining output frequencies of non linear systems subject to an input the frequency components of which are located in two separate frequency intervals and a new procedure for the implementation of the filter design the results achieved provide an effective approach to the solution to a more general etf design problem

artificial compressibility characteristics based schemes for variable density incompressible multispecies flows part ii multigrid implementation and numerical tests the paper presents an investigation of the accuracy and efficiency of artificial compressibility characteristics based cb schemes for variable density incompressible flows the cb schemes have been implemented in conjunction with a multigrid method for accelerating numerical convergence and a fourth order explicit runge kutta method for the integration of the governing equations in time the implementation of the cb schemes is obtained in conjunction with first second and third order interpolation formulas for calculating the variables at the cell faces of the computational volume the accuracy and efficiency of the schemes are examined against analytical and experimental results for diffusion broadening in two and three dimensional microfluidic channels a problem that has motivated the development of the present methods moreover unsteady inviscid simulations have been performed for variable density mixing layer the computations revealed that accuracy and efficiency depend on the cb scheme design the best multigrid convergence rates were exhibited by the conservative cb scheme which is obtained by the fully conservative formulation of the variable density incompressible equations

artificial compressibility characteristics based schemes for variable density incompressible multi species flows part i derivation of different formulations and constant density limit the paper presents various formulations of characteristics based schemes in the framework of the artificial compressibility method for variable density incompressible flows in contrast to constant density incompressible flows where the characteristics based variables reconstruction leads to a single formulation in the case of variable density flows three different schemes can be obtained henceforth labeled as transport conservative and hybrid schemes the conservative scheme results in pseudo compressibility terms in the multi species density reconstruction it is shown that in the limit of constant density the transport scheme becomes the original characteristics based scheme for incompressible flows but the conservative and hybrid schemes lead to a new characteristics based variant for constant density flows the characteristics based schemes are combined with second and third order interpolation for increasing the computational accuracy locally at the cell faces of the control volume numerical experiments for constant density flows reveal that all the characteristics based schemes result in the same flow solution but they exhibit different convergence behavior the multigrid implementation and numerical studies for variable density flows are presented in part ii of this study

implicit fully mesh less method for compressible viscous flow calculations a dual time implicit mesh less scheme is developed for solution of governing viscous flow equations the computational efficiency of the method is enhanced by adopting accelerating techniques such as local time stepping and residual smoothing the taylor series least square method is used for approximation of derivatives at each node which leads to a central difference spatial discretization the capabilities of the method are demonstrated by flow computations about standard cases at subsonic and transonic laminar flow conditions results are presented which indicate good agreements with the alternative numerical and experimental data the computational time is considerably reduced when using the proposed mesh less method compared with the explicit mesh less and finite volume counterparts using the same distribution of points

a second order time accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids a new second order time accurate fractional step method for solving unsteady incompressible navier stokes equations on hybrid unstructured grids is presented the nonstaggered grid method originally developed by rhie and chow 1983 aiaa j 21 1525 for steady flow and further extended by zang et al 1994 j comput phys 114 18 to unsteady flow on structured grids is employed in the present study to enforce mass conservation on hybrid unstructured grids the pressure and cartesian velocity components are defined at the center of each cell while the face normal velocities are defined at the mid points of the corresponding cell faces a second order fully implicit time advancement scheme is used for time integration and the resulting nonlinear equations are linearized without losing the overall time accuracy both the momentum and poisson equations are integrated with the finite volume method and the flow variables at the cell face are obtained using an interpolation scheme independent of cell shape the present numerical method is applied to four different benchmark problems and proves to be accurate and efficient

on the convexity of the carrying simplex of planar lotka volterra competitive systems in this paper we prove that a result by zeeman and zeeman on the convexity of carrying simplices in competitive lotka volterra systems differential equations dynamical systems and control science a festschift in honor to laurence markus 1994 pp 353 364 is partially wrong and we find necessary and sufficient conditions for which the carrying simplex of a planar competitive lotka volterra system is twice continuously differentiable in particular the main result by mierczynski j differential equations 111 1994 385 409 is sharp

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

fast multigrid method for solving incompressible hydrodynamic problems with free surfaces we develop of a finite volume multigrid euler scheme for solving three dimensional fully nonlinear ship wave problems the flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow the evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged the method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow a multigrid algorithm is used to accelerate convergence to a steady state

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

analysis of output frequencies of nonlinear systems in this paper an algorithm is derived for the determination of the output frequency ranges of nonlinear systems which extends previous results on the output frequencies of nonlinear systems to a more general situation the new results are significant for the analysis of the output frequency response of a wide class of nonlinear systems

output frequency response characteristics of nonlinear systems part ii overlapping effects and commensurate multi tone excitations the output response of nonlinear systems is strongly dependent on the extent to which different input frequency combinations intermodulations overlap one another and contribute to a common output frequency component this makes it hard to determine the actual number of output frequencies or to determine which of the many possible combinations contribute to a specified output frequency of interest systematic algorithms are presented for resolving these issues in a manner which is both efficient and straightforward to implement on a computer three broad classes of input are considered multi tone but nominally incommensurate sinusoids multi tone harmonically related sinusoids and multi tone excitations that are equi spaced in frequency the considered inputs also admit the possibility of a dc bias component since this can significantly affect the system response the algorithms are therefore applicable to a broad range of practical inputs and their use and benefits are illustrated with several examples

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

a fast pressure correction method for incompressible flows over curved walls abstract we have developed an explicit and direct pressure correction method for simulating incompressible flows over curved walls in order to integrate the navier stokes ns equations in time we have developed an explicit three stage third order runge kutta based projection method fastrk3 which requires solving the poisson equation for pressure only once per time step we have chosen to discretize the incompressible ns equations written in the orthogonal coordinates rather than the general formulation in curvilinear coordinates because the former does not contain cross derivatives in the advection diffusion laplacian and gradient operators thus the computational cost of solving the ns equations is substantially reduced and the numerical stencils of the finite difference approximations to these operators mirror that of the cartesian formulation this property also allows us to develop an fft based poisson solver for pressure fastpoc for those cases where the components of the metric tensor are independent of one spatial direction surfaces of linear translation e g curved ramps and bumps and surfaces of revolution e g axisymmetric ramps we have verified and validated fastrk3 and we have applied fastrk3 for simulating separated flows over ramps and a bump finally our results show that the new fft based poisson solver fastpoc is thirty to sixty times faster than the multigrid based linear solver depending on the tolerance set for the multigrid solver and the new flow solver fastrk3 is overall four to seven times faster when using fastpoc rather than multigrid in summary given that the computational mesh satisfies the property of orthogonality fastrk3 can simulate flows over curved walls with second order accuracy in space

a fast lattice green s function method for solving viscous incompressible flows on unbounded domains a computationally efficient method for solving three dimensional viscous incompressible flows on unbounded domains is presented the method formally discretizes the incompressible navier stokes equations on an unbounded staggered cartesian grid operations are limited to a finite computational domain through a lattice green s function technique this technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators the differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half explicit runge kutta scheme for the convective term a projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme linear complexity fast computation rates and parallel scalability are achieved using recently developed fast multipole methods for difference equations the accuracy and physical fidelity of solutions are verified through numerical simulations of vortex rings

output frequency response characteristics of nonlinear systems part i general multi tone inputs a systematic algorithm for determining the output frequencies generated by a nonlinear system subject to a general multi tone input including a bias input is presented the algorithm is very efficient and circumvents many of the difficulties of duplicate frequencies that are often generated using previous methods the inclusion of the dc bias component in the analysis is also important because of the significant effect that non zero mean inputs can have on nonlinear system response the algorithm is straightforward to implement on a computer and the results are illustrated by means of an example

segregated runge kutta methods for the incompressible navier stokes equations in this work we propose runge kutta time integration schemes for the incompressible navier stokes equations with two salient properties first velocity and pressure computations are segregated at the time integration level without the need to perform additional fractional step techniques that spoil high orders of accuracy second the proposed methods keep the same order of accuracy for both velocities and pressures the segregated runge kutta methods are motivated as an implicit explicit runge kutta time integration of the projected navier stokes system onto the discrete divergence free space and its re statement in a velocity pressure setting using a discrete pressure poisson equation we have analysed the preservation of the discrete divergence constraint for segregated runge kutta methods and their relation in their fully explicit version with existing half explicit methods we have performed a detailed numerical experimentation for a wide set of schemes from first to third order including implicit and imex integration of viscous and convective terms for incompressible laminar and turbulent flows further segregated runge kutta schemes with adaptive time stepping are proposed

segregated runge kutta time integration of convection stabilized mixed finite element schemes for wall unresolved les of incompressible flows in this work we develop a high performance numerical framework for the large eddy simulation les of incompressible flows the spatial discretization of the nonlinear system is carried out using mixed finite element fe schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint these additional terms introduced at the discrete level have been proved to act as implicit les models in order to perform meaningful wall unresolved simulations we consider a weak imposition of the boundary conditions using a nitsche s type scheme where the tangential component penalty term is designed to act as a wall law next segregated runge kutta srk schemes recently proposed by the authors for laminar flow problems are applied to the les simulation of turbulent flows by the introduction of a penalty term on the trace of the acceleration these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms srk schemes are excellent for large scale simulations since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level leading to two uncoupled systems the pressure system is a darcy type problem that can easily be preconditioned using a traditional block preconditioning scheme that only requires a poisson solver at the end only coercive systems have to be solved which can be effectively preconditioned by multilevel domain decomposition schemes which are both optimal and scalable the framework is applied to the taylor green and turbulent channel flow benchmarks in order to prove the accuracy of the convection stabilized mixed fes as les models and srk time integrators the scalability of the preconditioning techniques in space only has also been proven for one step of the srk scheme for the taylor green flow using uniform meshes moreover a turbulent flow around a naca profile is solved to show the applicability of the proposed algorithms for a realistic problem

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 characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids abstract this paper presents a new unstructured grid upwind finite volume algorithm for accurate numerical simulation of incompressible flows and convection heat transfer on unstructured grids it is an upwind method at both the differential equation level and discretized equation level based on the method of characteristics this is made possible with the introduction of chorin s j comput phys 2 1967 12 26 artificial compressibility formulation to the incompressible flow equations flow variables are calculated along characteristics and their initial values are interpolated based on the signs of the corresponding characteristic speed in addition an upwind biased interpolation method of third order accuracy is used for interpolating flow variables on unstructured grids with these inherent upwinding techniques for evaluating convection fluxes at control volume surfaces no artificial viscosity is required and the method is accurate and less sensitive to the grid orientation thus suitable for unstructured grid application the use of the finite volume method combined with unstructured grids makes the scheme very flexible in dealing with very complex boundary geometries while remaining to be fully conservative at the cell as well as global level the discretized equations are solved by an explicit multistage runge kutta time stepping scheme which is found to be efficient in terms of cpu and memory overheads a computer code has been developed which uses the numerical methods presented and a highly efficient edge based data structure for flux calculation and data storage a number of well documented test cases including 2d laminar flow in a channel with a backward facing step forced and natural convection flows have been calculated in order to validate the code and evaluate the performance of the numerical algorithm good agreement between the computed and measured results have been obtained for all the cases to demonstrate the accuracy of the third order scheme flow in another channel with a backward facing step studied by gartling int j numer meth fluids 1990 11 is simulated with both the third order and first order schemes it is shown that the third order scheme can produce a velocity profile across the channel that has perfect agreement with that of gartling local citation whilst the first order scheme cannot even predict the separation near the upper wall the proposed numerical methods are also used to calculate 3d incompressible inviscid flows and results show that the method is accurate and robust for general 3d simulation the 3d navier stokes code is later used to study a lid driven cavity flow at a reynolds number of 400 the unstructured grid used has 97 336 nodes and 455 625 tetrahedral elements with strong grid clustering near the walls the convergence rate is found to be satisfactory which shows that the method is also robust for 3d navier stokes simulation the results obtained agree well with those reported by jiang et al comput meth appl mech eng 114 109 1994 results obtained using the second order central scheme with artificial viscosity are also presented for comparison it is observed that the third order characteristics scheme is more accurate than the latter thus confirming its third order accuracy

an efficient method for temporal integration of the navier stokes equations in confined axisymmetric geometries a method for temporal integration of the navier stokes equations written in cylindrical coordinates is described the objective is to avoid the severe time step limitation usually encountered in confined axisymmetric geometries e g pipe flow caused by a fine azimuthal grid spacing around the centerline and the desire to refine the grid in the radial direction near walls avoiding severe time step limitations usually involves treating all terms with derivatives in the radial and azimuthal directions with an implicit time integration scheme however this leads to a set of coupled nonlinear equations which generally require complex and costly solution procedures the scheme described in this paper decomposes the computational domain into two regions within each region only the derivatives in one coordinate direction is treated implicitly conditions at the interface between the regions are determined to maintain the overall temporal accuracy of the basic time integration schemes results from a direct numerical simulation dns of turbulent pipe flow are validated against computational and experimental results from the literature it is demonstrated that this new scheme allows for larger time steps than other schemes leading to significant cpu savings

a lagged implicit segregated data reconstruction procedure to treat open boundaries the problem of treating open boundaries is still a challenging one applying fully developed condition is constrained to long enough domains without having enough physical evidence about what happens on boundaries the domain extent could not be shortened and computational costs could not be reduced from the advent of free open boundary conditions they were confined to mixed finite element procedures recent works have extended their application to coupled finite volume solvers based on the shape function data reconstruction a wider class of flow solvers available however rely on the segregated procedure where the velocity components and pressure are solved in succession moreover many finite volume algorithms do not use the shape function reconstruction in this work by proposing a lagged implicit procedure we have extended the application of the open boundary condition to these wider classes of flow solvers the proposed extension is a combination of lagged implicit data reconstruction and overall mass conservation enforcement which is easily applicable to any segregated and coupled flow solver to validate the compatibility of this extension benchmark problem of backward facing step is solved on successively truncated domains where open boundary may pass through recirculation zones results show that the proposed extension works fine for that problem it reduced the computational domain length and hence memory by a factor of 4 6 and the required computational time by a factor of 21 flow passing a cylinder is also solved which proves that the method could be applied to external flow problems as well

a nonreflecting outlet boundary condition for incompressible unsteady navier stokes calculations abstract the goal of this work is to adapt a nonreflecting outlet boundary condition derived from a wave equation to the numerical solution of the full incompressible navier stokes equations for an elliptic unsteady free shear flow the numerical results show that a significant improvement is achieved with this nonreflecting boundary condition in comparison with the results obtained by using free boundary layer type conditions the physical phenomena studied concern the onset of the kelvin helmholtz instability in the free non forced shear layer and certain 2d characteristics of transition towards turbulence these phenomena are simulated naturally without imposing perturbations the frequency of the organized vortices and the spread of the mixing layer are correctly predicted the performances of the method are shown through comparison with the physical experiments owing to the nonreflecting boundary conditions the feedback noises are inhibited effectively so that the computation domain can be reduced and the dynamic characteristics of the flow are maintained up clearly to the outlet boundary

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

baseband volterra filters for implementing carrier based nonlinearities a diagonal coordinate representation for volterra filters is developed and exploited to derive efficient volterra filter implementations for processing carrier based input signals in the diagonal coordinate representation the output is expressed as a sum of linear filters applied to modified input signals hence linear filtering methods are employed to implement the nonlinear filter on a baseband version of the input downsampling is then used to reduce computational complexity the same approach is employed to develop efficient implementations for processing continuous time carrier based signals pulse amplitude modulated signals and frequency division multiplexed input signals

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

family of runge kutta based algorithms for unsteady incompressible flows a family of algorithms based on implicit runge kutta time integration suitable for unsteady incompressible flow is developed spatially discretized momentum equations are integrated in time using i

mapping nonlinear integro differential equations to a generalized describing function form an algebraic relationship is derived which enables the generalized amplitude dependent describing function to be expressed in terms of the parameters of a general nonlinear integro differential equation the algorithm is not restricted to a specific input but treats the whole class of harmonically related phase independent sinusoidal inputs and is illustrated by a number of examples

practical frequency response analysis of non linear time delayed differential or difference equation models a systematic algorithm for generating the polyharmonic balance equations for any system within a broad class of time delayed differential or non linear difference equation models is presented the method which is readily automated enables the balance equations to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response or amplitude dependent describing function is then readily computed the method is illustrated by means of examples including both a time delayed differential system example and a discrete time narx model application the results are validated against detailed numeric simulation which confirms the accuracy and efficacy of the approach

low storage implicit explicit runge kutta schemes for the simulation of stiff high dimensional ode systems implicit explicit imex runge kutta rk schemes are effective for time marching ode systems with both stiff and nonstiff terms on the rhs such schemes implement an often a stable or better implicit rk scheme for the stiff part of the ode which is often linear and simultaneously a more convenient explicit rk scheme for the nonstiff part of the ode which is often nonlinear low storage rk schemes are especially effective for time marching high dimensional ode discretizations of pde systems on modern cache based computational hardware in which memory management is often the most significant computational bottleneck in this paper we develop and characterize eight new low storage implicit explicit rk schemes which have higher accuracy and better stability properties than the only low storage implicit explicit rk scheme available previously the venerable second order crank nicolson runge kutta wray cn rkw3 algorithm that has dominated the dns les literature for the last 25 years while requiring similar storage two three or four registers of length n and comparable floating point operations per timestep

convergence to equilibrium on invariant d hypersurfaces for strongly increasing discrete time semigroups abstract convergence to equilibrium for every positive semi orbit of strongly increasing discrete time semigroups on strongly ordered spaces is proved the domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered banach space which is continuously imbedded into a banach lattice the semigroup is assumed to be order compact and every positive semi orbit is assumed to be order bounded the crucial hypothesis is the ljapunov stability of all equilibria it is also proved that the set of equilibria is a simply ordered arc the key tools are invariant d hypersurfaces

mapping from parametric characteristics to generalized frequency response functions of non linear systems based on the parametric characteristic of the nth order generalized frequency response function gfrf for non linear systems described by a non linear differential equation nde model a mapping function from the parametric characteristics to the gfrfs is established by which the nth order gfrf can be directly written into a more straightforward and meaningful form in terms of the first order gfrf i e an n degree polynomial function of the first order gfrf the new expression has no recursive relationship between different order gfrfs and demonstrates some new properties of the gfrfs which can explicitly unveil the linear and non linear factors included in the gfrfs and reveal clearly the relationship between the nth order gfrf and its parametric characteristic as well as the relationship between the nth order gfrf and the first order gfrf the new results provide a novel and useful insight into the frequency domain analysis and design of non linear systems based on the gfrfs several examples are given to illustrate the theoretical results

boundary conditions for open boundaries for the incompressible navier stokes equation abstract in this paper we investigate new boundary conditions for the incompressible time dependent navier stokes equation especially inflow and outflow conditions are considered the equations are linearized around a constant flow so that we can use laplace fourier technique to investigate the strength of boundary layers at open boundaries such layers are unphysical and new boundary conditions are proposed so that these boundary layers are suppressed we also show that the boundary conditions we propose do not produce divergence furthermore they give solutions that do not grow in time as long as the forcing function in the system does not we also discuss the numerical treatment of boundaries when fourth order accurate finite difference operators are used to approximate spatial derivatives using higher order methods introduces eigensolutions with boundary layer thickness of the same order of magnitude as the grid size these eigensolutions have to be suppressed in order to not destroy the fourth order accuracy of the method numerical results for the non linear navier stokes equation together with the new boundary conditions are presented these calculations confirm that the results for the linearized problem hold for the non linear problem as well

manifolds of balance in planar ecological systems abstract in the classic 2 species lotka volterra competition model and more general competitive planar kolmogorov models there is a continuous curve called the carrying simplex that links all non zero steady states and attracts all non zero population densities this curve is where the opposing processes of population growth and decline balance in this paper we use stability analysis and index theory to show that such a curve also exists when the interactions between two species are more general such as co operative or predator prey provided that reasonable biologically motivated conditions hold for example both species experience intraspecific competition and all population densities remain bounded for all time we consider systems where there is at most one co existence steady state the balance manifold is formed of heteroclinic orbits and attracts all non zero population densities but unlike its competitive analogue the curve is no longer necessarily continuously differentiable

high order finite difference and multigrid methods for spatially evolving instability in a planar channel a computational study of the spatial instability of planar poiseuille flow is presented a fourth order finite difference with a fully implicit time marching scheme is developed on a staggered grid a semi coarsening multigrid method is applied to accelerate convergence for the implicit scheme at each time step and a line distributive relaxation is developed as a fast solver which is very robust and efficient for anisotropic grids a new treatment for outflow boundary conditions makes the buffer area as short as one wavelength the computational results demonstrate high accuracy in terms of agreement with linear theory and excellent efficiency in the sense that cost is comparable to and usually less than explicit schemes

preconditioned multigrid methods for unsteady incompressible flows a highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3d steady and time dependent incompressible flows thek turbulence model is used to estimate the effects of turbulence the model equations are solved together with the n s equations in a strongly coupled way and acceleration techniques like the multigrid method are also used for the turbulence model equations for unsteady problems a dual time stepping procedure is adopted to satisfy the divergence free constraint and to obtain a time accurate solution to improve the performance of this approach for small physical time steps a modification to residual smoothing parameters is proposed the numerical algorithm and the turbulence model are validated first by calculating unsteady inviscid flow around an oscillating cylinder unsteady laminar flow past a circular cylinder and steady high reynolds number turbulent flow over a 6 1 prolate spheroid then the three dimensional time dependent turbulent flow over a spheroid when it is undergoing a pitch up maneuver is calculated and compared with experimental data

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

efficient and non reflecting far field boundary conditions for incompressible flow calculations abstract traditionally the artificial compressibility ac method of chorin is used for simulation of low speed and incompressible flows in this method the difficulty of continuity and momentum equations decoupling is removed by adding an artificial time derivative of pressure to the continuity equation for the first time a fully two dimensional upwind scheme was presented for ac equations by using characteristic structure of equations by the authors in this paper a new remote boundary calculation method is presented by using the idea of characteristics for these equations instead of simple boundary conditions which usually are employed for incompressible flows the flow quantities at the far field boundaries are evaluated by compatibility relations of characteristic equations this is implemented by assuming a row of ghost cells outside of the far field boundary and using the flux calculation method based on characteristics similar to the inside of computational domain the idea in conjunction with multidimensional characteristic based scheme was tested for incompressible flow around circular cylinder in comparison with conventional far field boundary condition and showed good improvements in the terms of accuracy and convergence speed

automatic computation of polyharmonic balance equations for non linear differential systems an automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross product non linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response in the form of a multi input amplitude dependent describing function is therefore readily computed the method is illustrated by means of an example and the results validated against detailed numeric simulation

smoothness of the carrying simplex for discrete time competitive dynamical systems a characterization of neat embedding abstract the paper is concerned with the question of smoothness of the carrying simplex s for a discrete time dissipative competitive dynamical system we give a necessary and sufficient criterion for s being a c 1 submanifold with corners neatly embedded in the nonnegative orthant formulated in terms of inequalities between lyapunov exponents for ergodic measures supported on the boundary of the orthant this completes one thread of investigation occasioned by a question posed by m w hirsch in 1988 besides amenable conditions are presented to guarantee the c r r 1 smoothness of s in the time periodic competitive kolmogorov systems of odes examples are also presented one in which s is of class c 1 but not neatly embedded the other in which s is not of class c 1

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

describing functions volterra series and the analysis of non linear systems in the frequency domain a relationship between the higher order frequency response functions of the volterra series model and the well known describing function representation for non linear systems is derived it is shown that a large class of generalized describing functions can be derived from the volterra model and that this approach removes the restriction to specific inputs the concept of a worst case transfer function is introduced and examples are included to illustrate the results

finite volume compact schemes on staggered grids compact finite difference schemes have been recently used in several direct numerical simulations of turbulent flows since they can achieve high order accuracy and high resolution without exceedingly increasing the size of the computational stencil the development of compact finite volume schemes is more revolved due to the appearance of surface and volume integrals while pereira et al j comput phys 167 2001 and smirnov et al aiaa paper 2546 2001 focused on collocated grids in this paper we use the staggered grid arrangement compact schemes can be tuned to achieve very high resolution for a given formal order of accuracy we develop and test high resolution schemes by following a procedure proposed by lele j comput phys 103 1992 which to the best of our knowledge has not yet been applied to compact finite volume methods on staggered grids results from several one and two dimensional simulations for the scalar transport and navier stokes equations are presented showing that the proposed method is capable to accurately reproduce complex steady and unsteady flows

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

a nonlinear multigrid method for the three dimensional incompressible navier stokes equations a nonlinear multigrid method is developed for solving the three dimensional navier stokes equations in conjunction with the artificial compressibility formulation the method is based on the full multigrid fmg full approximation storage fas algorithm and is realized via an unsteady type procedure according to which the equations are not solved exactly on the coarsest grid but some pseudo time iterations are performed on the finer grids and some on the coarsest grid the multigrid method is implemented in conjunction with a third order upwind characteristics based scheme for the discretization of the convection terms and the fourth order runge kutta scheme for time integration the performance of the method is investigated for three dimensional flows in straight and curved channels as well as flow in a cubic cavity the multigrid acceleration is assessed in contrast to the single grid and mesh sequencing algorithms the effects of various multigrid components on the convergence acceleration such as prolongation operators as well as pre and postrelaxation iterations are also investigated

an improvement of fractional step methods for the incompressible navier stokes equations abstract a numerical method for computing three dimensional unsteady incompressible flows is presented the method is a predictor corrector technique combined with a fractional step method each time step is advanced in three sub steps the novel feature of the present scheme is that the poisson equation for the pressure is solved only at the final sub step resulting in substantial savings in computing time it is shown that the method allows a larger cfl number and reduces the computing cost without loss of accuracy by satisfying the continuity equation only at the last sub step numerical solutions for the decaying vortices and flow over a backward facing step are obtained and compared with analytical and other numerical results

interpretation of non linear frequency response functions analytical and graphical methods of interpreting generalised frequency response functions for non linear systems are derived it is shown that non linear phenomena can be classified into intra kernel and inter kernel interference and that worst case responses can be computed the results are illustrated using several discrete and continuous time non linear systems

robust feedback synthesis for nonlinear integro differential equation models using generalized describing functions we propose a technique for synthesizing robust feedback systems for plants described by nonlinear integro differential equations nide with considerable coefficient uncertainties the technique is based on the concept of generalized frequency response functions gfrf and exploits recent results that enable ready derivation of these from the nide description of the plant a powerful synthesis procedure that combines the gfrf and quantitative feedback theory qft technique is given the proposed procedure has several advantages over the existing nonlinear qft methods a nonlinear chemical reactor example is solved using the procedure and is found to yield satisfactory results

an investigation into the characteristics of non linear frequency response functions part 2 new analysis methods based on symbolic expansions and graphical techniques in part 1 of this paper the concepts of input and output frequency subdomains were introduced to give insight into the relation between one dimensional and multi dimensional frequency spaces the visualisation of both magnitude and phase responses of third order generalized frequency response functions was also presented in this the second part symbolic expansion techniques are introduced and together with the results achieved in part 1 yield new methods for analysing the properties of generalized frequency response functions case studies are included to illustrate the application of the new methods

parallel multigrid computation of unsteady incompressible viscous flows using a matrix free implicit method and high resolution characteristics based scheme abstract a three dimensional parallel unstructured non nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated the finite volume navier stokes solver is based on the artificial compressibility approach with a high resolution method of characteristics based scheme for handling convection terms the unsteady flow is calculated with a matrix free implicit dual time stepping scheme the parallelization of the multigrid solver is achieved by multigrid domain decomposition approach mg dd using single program multiple data spmd and multiple instruction multiple data mimd programming paradigm there are two parallelization strategies proposed in this work first strategy is a one level parallelization strategy using geometric domain decomposition technique alone second strategy is a two level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques message passing interface mpi and openmp standard are used to communicate data between processors and decompose loop iterations arrays respectively the parallel multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid driven cavity flow a maximum speedup of 22 5 could be achieved on 32 processors for instance the lid driven cavity flow of re 1000 the results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements a detailed study of the time step size and number of pseudo sub iterations per time step required for simulating unsteady flow are presented in this paper

an exclusion principle for periodic competitive systems in three dimensions an exclusion principle for periodic competitive systems in three dimensionst rafael ortega and antonio tineo departamento de matemzitica aplicada universidad de granada 18071 granada spain and departmento de matem iticas facultad de ciencias universidad de los andes 5101 m6rida venezuela received 29 february 1996 received in revised form 25 september 1996 received for publication 18 february 1997 key words and phrases logistic equation competing species problem periodic solutions exclusion principle 0 introduction in this paper we consider the following three dimensional periodic system u uifi t ul u2 u u i 0 1 0 and uif t u is locally lipschitz continuous in u we shall assume that the system is strongly competitive and dissipative and prove the following exclusion principle theorem 0 1 suppose that a f t ul 2

an accurate compact treatment of pressure for colocated variables we propose a new scheme for the pressure treatment in computations of incompressible flow using a colocated grid arrangement to avoid oscillations associated with the sparse non compact stencil we introduce a compact fourth order equivalent of this stencil and study its advantages over the classical second order averaging procedure

linearization and invariant manifolds on the carrying simplex for competitive maps abstract a result due to m w hirsch states that most competitive maps admit a carrying simplex i e an invariant hypersurface of codimension one which attracts all nontrivial orbits the common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex however this manifold is normally non smooth therefore not every tool coming from differential geometry can be applied in this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo unstable manifold by an invariant foliation this implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex we further construct the stable and unstable manifolds on the carrying simplex our results give partial responses to hirsch s problem regarding the smoothness of the carrying simplex we discuss some applications in classical models of population dynamics

multidimensional upwinding for incompressible flows based on characteristics in this paper the multidimensional characteristic based upwind scheme mcb which has been recently introduced by the authors is applied to two another benchmark problems namely flow in a channel with a backward facing step and two dimensional steady and unsteady flows past a circular cylinder extension of mcb scheme for calculating convective fluxes on non cartesian grids is presented here for the flow over backward facing step obtained results were compared against well known experimental data and the results show high accuracy of mcb scheme and faster convergence rate with respect to conventional cb scheme in the case of flow over circular cylinder the flow at steady and transient regimes is investigated by mcb scheme again the results obtained by mcb are compared to the other results in the literature and show good agreement with them also rapid convergence rate of mcb was observed in this case too it is concluded that the genuinely multidimensional characteristic based mcb scheme has been introduced earlier by the authors is a robust and powerful scheme for modeling incompressible viscous flows for achieving the high accuracy and remarkable advantage in convergence rate with respect to conventional characteristic based schemes

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples