resolution of the 2d navier stokes equations in velocity vorticity form by means of an influence matrix technique abstract an influence matrix technique is proposed to enforce both the continuity equation and the definition of the vorticity in the treatment of the 2d incompressible navier stokes equations it is shown and supported by numerical experiments that at each time step the divergence is actually equal to zero within machine accuracy the same result is obtained for the definition of the vorticity

efficient adaptive cartesian vorticity transport solver for vortex dominated flows an efficient solver for the velocity vorticity form of the navier stokes equations on adaptive cartesian grids is presented the excessive numerical dissipation common to most grid based navier stokes solvers is avoided by solving the fluid dynamic equations in vorticity conservation form additionally an adaptive cartesian solver is employed to efficiently capture and preserve vorticity on the fly as the flow develops for practical purposes this solver would be used in the wake region and coupled with a full navier stokes solver in the near body region thus allowing vorticity to be accurately generated and then convected in the wake region with minimal dissipation the adaptive cartesian framework allows for the rapid evaluation of the velocity field using a fast summation technique based on the cartesian treecode method the implementations of both the solution algorithm and velocity calculation are described in detail results are presented for vortex convection applications that show good agreement with the analytical solution and the accuracy of the scheme is verified numerically using a series of increasingly fine grids additionally fully three dimensional flow in the presence of a vortex ring is investigated and results are shown to be in very close agreement with published data

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

bicoprime factor stability criteria for linear time varying systems in the nest algebra this paper presents a bicoprime factorisation approach to deal with various stability problems for linear discrete time varying systems within the framework of nest algebra first we show that a s

a spectral solver for the navier stokes equations in the velocity vorticity formulation for flows with two nonperiodic directions a novel pseudospectral solution procedure for the navier stokes equations in the velocity vorticity formulation suitable to simulate flows with two nonperiodic directions is proposed an influence matrix method including a tau correction procedure has been employed to elicit an algorithmic substitute for thea priorilacking boundary conditions for the vorticity following o daube j comput phys 103 402 1992 the influence matrix is built by enforcing either the definition of the vorticity or the continuity equation at the boundary of the domain the influence matrix is nonsingular in both cases the order of the influence matrix is twice larger than one would expect from analogous solution methods based on finite differences or finite elements the spatial discretization is based on a 2d chebyshev expansion on anonstaggeredgrid of collocation points and the boundary conditions are imposed via the lanczos tau procedure the time marching scheme is adams bashforth for the advection term and crank nicolson for the viscous term the proposed scheme yields machine accurate divergence free flow fields and the definition of the vorticity is satisfied within machine accuracy

solution of the div curl problem in generalized curvilinear coordinates we propose a method for solving the div curl problem on a structured nonorthogonal curvilinear grid the differential operators are discretized using a mac scheme for the unknowns in such a way that the discrete counterparts of the usual vector analysis relations are satisfied the derived discrete problem is then solved by performing a helmholtz type decomposition of the unknown vector field this allows us to obtain a vector field for which both divergence and curl are satisfied to within machine accuracy the method is validated for several configurations in two and three dimensions and its accuracy is numerically checked

a comparative study of finite element and finite difference methods for cauchy riemann type equations a least squares formulation of the system operatorname div bf u rho operatorname curl bf u zeta is surveyed from the viewpoint of both finite element and finite difference

stability analysis for time varying systems via quadratic constraints abstract in this paper we introduce the concept of quadratic constraint qc to deal with various stability problems for infinite dimensional linear discrete time varying ltv systems first we derive a necessary and sufficient condition for the close loop stability based on quadratic constraints qcs an equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system moreover applying this stability criterion and the complete finiteness of the nest algebra we show that the plant is stabilizable if and only if it has a single strong representation finally we use qcs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric

a separable frechet space of almost universal disposition abstract the gurari space is the unique separable banach space g which is of almost universal disposition for finite dimensional banach spaces which means that for every e 0 for all finite dimensional normed spaces e f for every isometric embedding e e g there exists an e isometric embedding f f g such that f e e we show that g n with a special sequence of semi norms is of almost universal disposition for finite dimensional graded frechet spaces the construction relies heavily on the universal operator on the gurari space recently constructed by garbulinska wegrzyn and the third author in addition we consider a non graded sequence of semi norms on g n with which the space g n is of almost universal disposition for finite dimensional frechet spaces with a fixed sequence of semi norms in both cases this yields in particular that g n is universal in the class of all separable frechet spaces

topological stable rank of nest algebras we establish a general result about extending a right invertible row over a banach algebra to an invertible matrix this is applied to the computation of right topological stable rank of a split exact sequence we also introduce a quantitative measure of stable rank these results are applied to compute the right left topological stable rank for all nest algebras this value is either 2 or infinity and rtsr t n 2 occurs only when n is of ordinal type less than 2 and the dimensions of the atoms grows sufficiently quickly we introduce general results on partial matrix algebras over a banach algebra this is used to obtain an inequality akin to rieffel s formula for matrix algebras over a banach algebra this is used to give further insight into the nest case

a finite element method for diffusion dominated unsteady viscous flows a general conforming finite element scheme for computing viscous flows is presented which is of second order accuracy in space and time viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm a method for solving the algebraic equations at each time step is given the method is demonstrated on two test problems one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step

a numerical study of the two dimensional navier stokes equations in vorticity velocity variables abstract solution methods for compact finite difference schemes are applied to the vorticity velocity form of the two dimensional unsteady navier stokes equations numerical experiments for stagnation point and driven cavity flows are described

stabilizability and existence of system representations for discrete time time varying systems in this paper right and left representations as an alternate but equivalent framework to coprime factorizations of operators are developed the main theorem of the paper establishes that a linear time varying discrete time plant is stabilizable if and only if its graph can be represented as the range respectively kernel of a causal bounded operator which is left respectively right invertible the proof relies on certain factorization theorems of arveson for nest algebras the paper extends the youla parametrization of all stabilizing compensators to this framework also it is proven that a time invariant plant that is not stabilizable by a time invariant compensator is not stabilizable with a time varying compensator an example of a time varying plant of feintuch is considered and shown to be not stabilizable finally the continuous time case is examined and the problems encountered in extending the proof are discussed however it is shown that a stabilizable plant must have a closed graph

splitting of exact sequences of frechet spaces in the absence of continuous norms abstract in the present paper we study the splitting of a short exact sequence 0 g e s f 0 of nuclear frechet spaces where f need not admit a continuous norm i e e or f may be spaces of continuous functions on an open subset of r n or on a compact c manifold if e is such a space then we give a necessary and sufficient condition for the splitting of the sequence in terms of a condition on s and of linear topological invariants loc and dnloc this is used to give a structure theory of the space s n a space which is isomorphic to many spaces of smooth functions and to study the splitting of differential complexes the paper offers a different approach to problems which have been studied in domanski vogt j funct anal 153 1998 203

the numerical solution of the navier stokes equations for 3 dimensional unsteady incompressible flows by compact schemes abstract this paper describes a numerical method for solving the navier stokes equations for unsteady incompressible 3 dimensional flows using velocity vorticity variables and irregular cartesian grids the method involves solving cauchy riemann type equations for the velocity and transport diffusion equations for the vorticity whose solenoidal vorticity components are obtained by solving a poisson equation for a suitably chosen scalar potential the importance of boundary conditions is examined the difference equations are solved by iterations in order to permit exploitation of parallel and vector computing methods numerical experiments confirm the second order spatial and temporal accuracy of the method

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

saddle connections near degenerate critical points in stokes flow within cavities streamline patterns and their bifurcations in two dimensional incompressible flow near a simple degenerate critical point away from boundaries are investigated by using the normal form theory for the streamfunction obtained by m brons j n hartnack streamline topologies near simple degenerate critical points in two dimensional flow away from boundaries phys fluids 11 1999 314 324 for the normal form of order six a bifurcation diagram is constructed with two bifurcation parameters the theory is applied to the patterns and bifurcations found numerically in the studies of stokes flow in a double lid driven rectangular cavity with two control parameters the cavity aspect ratio a height to width and the speed ratio s bifurcations in the cavity are obtained using an analytic solution for the streamfunction developed for any value of s and a using this solution for special values s a a global bifurcation is identified with a single heteroclinic connection which connects three saddle points in a triangle and does not appear in the unfolding of the simple linear degenerate critical points lying in a line y constant

unsteady solution of incompressible navier stokes equations abstract a new time accurate finite difference numerical method for solving incompressible navier stokes equations is presented with primitive variables as the unknowns the numerical scheme is a crank nicolson implicit treatment of all terms of the equations with central differencing for space derivatives the pressure and the nonlinearities in convection terms are not lagged to obtain the incompressible solution at the advanced time level a continuous auxiliary system is introduced in pseudo time with artificial compressibility time accurate solutions are presented for two dimensional fluid flows in a square cavity with an impulsively starting lid and with an oscillating lid

on the advantages of the vorticity velocity formulations of the equations of fluid dynamics the mathematical properties of the pressure velocity and vorticity velocity formulations of the equations of viscous flow are compared it is shown that a vorticity velocity formulation exists which has the interesting property that noninertial effects only enter into the problem through the implementation of initial and boundary conditions this valuable characteristic along with other advantages of the vorticity velocity approach are discussed in detail

the fast adaptive composite grid method fac algorithms for advanced computers fac is a fast systematic process for the numerical solution of partial differential equations on adapted grids this paper reports on numerical studies with fac in advancing computing environments including simulations with multiprocessor array architectures and actual tests on the hep we also analyze the parallel complexity of fac for general local grid applications

compact finite difference schemes for mixed initial boundary value problems this paper discusses a class of compact second order accurate finite difference equations for mixed initial boundary value problems for hyperbolic and convective diffusion equations convergence is proved by means of energy arguments and both types of equations are solved by similar algorithms for hyperbolic equations an extension of the lax wendroff method is described which incorporates dissipative boundary conditions upwind downwind differencing techniques arise as the formal hyperbolic limit of the convective diffusion equation finally a finite difference chain rule transforms the schemes from rectangular to quadrilateral subdomains

steady and unsteady incompressible flow in a double driven cavity using the artificial compressibility ac based characteristic based split cbs scheme in this paper the explicit characteristic based split cbs scheme has been employed to solve both steady and unsteady flows inside a non rectangular double driven cavity this problem is recently suggested as a benchmark problem for incompressible flows both unstructured and structured meshes have been employed in the present study to make sure that the predicted results are as close to reality as possible the results obtained show the existence of steady state at lower reynolds numbers 1000 and transient states at higher reynolds numbers the flow approaches a turbulent state as the reynolds number is increased to 10 000 copyright 2005 john wiley sons ltd

analysis of the cell vertex finite volume method for the cauchy riemann equations this paper initiates a study of finite volume methods for linear first order elliptic systems by performing a stability and convergence analysis of the cell vertex approximation of the cauchy riemann equations the approach is based on reformulating the scheme as a petrov galerkin finite element method with continuous bilinear trial functions and piecewise constant test functions optimal error bounds are derived in a mesh dependent norm and the counting problem which may occur due to geometry and boundary conditions is considered

on a generalization of the youla kucera parametrization part i the fractional ideal approach to siso systems abstract in this paper we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for siso systems within this mathematical framework we give necessary and sufficient conditions so that a plant is internally strongly bistably stabilizable or admits a weak coprime factorization moreover we show how to generalize the youla kucera parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization this parametrization is generally affine in two free parameters

on finite element approximation of the gradient for solution of poisson equation a nonconforming mixed finite element method is presented for approximation of w with w f w r 0 convergence of the order left nabla w u h right 0 omega mathcal o h 2 is proved when linear finite elements are used only the standard regularity assumption on triangulations is needed

graphs of time varying linear systems and coprime factorizations abstract the relationship between the system p having a right coprime factorization p ab 1 and the graph of p being represented as the range of the 2 1 column matrix with entries b and a is discussed for time varying systems

a unified numerical treatment of the wave equation and the cauchy riemann equations a unified second order accurate finite difference approach to treating the wave equation and the cauchy riemann equations is developed the procedure is motivated by an analysis of properties of weak solutions

a mass conserving vorticity velocity formulation with application to nonreacting and reacting flows in a commonly implemented version of the vorticity velocity formulation the governing equations for the fluid dynamics are expressed as two poisson like velocity equations together with the vorticity transport equation however for some flows with large vorticity gradients spurious mass loss or gain can be observed in order to conserve mass a modification to the vorticity velocity formulation is proposed involving the substitution of the kinematic definition of vorticity in certain terms of the fluid dynamic equations this modified formulation results in a broader computational stencil when the equations are in a second order accurate discretized form and a stronger coupling between the predicted vorticity and the curl of the predicted velocity field the resulting system of elliptic equations which includes the energy and species transport equations for the reacting flow case is discretized with finite differences on a nonstaggered grid and is then solved using newton s method both the unmodified and modified vorticity velocity formulations are applied to two problems with high vorticity gradients 1 incompressible axisymmetric fluid flow through a suddenly expanding pipe and 2 a confined axisymmetric laminar flame with detailed chemistry and multicomponent transport generated on a burner whose inner tube extends above the burner surface the modified formulation effectively eliminates the spurious mass loss in the two test cases to within an acceptable tolerance the two cases demonstrate the broader range of applicability of the modified formulation as compared with the unmodified formulation

stabilizability representations and factorizations for time varying linear systems abstract we study the system representations factorizations and stabilizability of a discrete time time varying linear system in the framework of nest algebra we shown that every time varying linear system admits a normalized left factorization while it may possibly not have a right factorization the relationship between the system representation and factorization is studied and some new stabilizability criteria are established

analysis of the cell vertex finite volume method for hyperbolic problems with variable coefficients an analysis of the cell vertex finite volume method is presented for a scalar linear hyperbolic equation with variable coefficients the cell vertex approximation is shown to be second order convergent in the l2 norm and first order convergent in the h1 norm provided the exact solution belongs to h3 and the characteristic curves of the differential equation are transversal to mesh lines on the closure of the domain of integration when characteristics are parallel to mesh lines on a subset of positive measure the scheme is only first order convergent in the l2 norm however if transversality fails on part of the outflow boundary only then the cell vertex scheme is second order convergent in a weighted l2 norm

a compact monotonic discretization scheme for solving second order vorticity velocity equations abstract this paper presents a numerical method for solving the steady state navier stokes equations for incompressible fluid flows using velocities and vorticity as working variables the method involves solving a second order differential equation for the velocity and a convection diffusion equation for the vorticity in cartesian grids the key to the success of the numerical simulation of this class of flow equations depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity in this paper we present a monotonic advection diffusion multi dimensional scheme and a theoretically rigorous implementation of vorticity boundary conditions while the derivation of the proposed integral vorticity boundary condition is more elaborate and is more difficult to solve than conventional local approaches the present approach offers significant advantages in this study both lid driven and backward facing step problems have been selected for comparison and validation purposes

a fast algorithm for simulation of a spatially evolving two dimensional planar mixing layer abstract an algorithm of exceptional computational efficiency is developed for numerical simulation of spatially evolving instability waves in an incompressible fluid the efficiency of the algorithm is demonstrated in a simulation of forced instability waves in a two dimensional planar mixing layer computationally analogous to the forced flow experiment of miksad the navier stokes equations in vorticity streamfunction form are solved on a rectangular domain with inflow outflow boundary conditions the finite difference algorithm exhibits 1 an asymptotic count of 2mnlog 2 m operations per discrete time step on an m n mesh comparable in computational efficiency to pseudospectral methods used to model temporally evolving instabilities 2 global 2nd order accuracy in time and space and 3 structure which is vectorizable for implementation on array processors execution rate on a cray x mp is 0 056 seconds per time step for a computational mesh of 256 144 resolution among the recent advances in numerical techniques which contribute to this efficiency are semi implicit temporal discretization approximate factorization of the matrix representation of the implicit diffusion operator and fast direct solution of poisson s equations moreover the algorithm is shown to extend naturally and efficiently to three spatial dimensions

transitivity in simultaneous stabilization for a family of time varying systems abstract this paper develops necessary and sufficient conditions for the so called transitivity and strong transitivity precise definitions later for a family of discrete time varying linear systems which when satisfied lead to characterizations for simultaneous stabilizability and simultaneously stabilizing feedback controllers some criteria for the strong transitivity are established in terms of single coprime factorizations of only one plant and one controller the resulting characterization involves coprime factorizations of fewer systems compared to previous work

coprime factorization and dynamic stabilization of transfer functions it is known that a matrix valued transfer function p has a stabilizing dynamic controller q i e begin smallmatrix i q p i end smallmatrix 1 in rm h infty iff p has a right or left coprime factorization we show that the same result is true in the operator valued case thus the standard youla bongiorno parameterization applies to every dynamically stabilizable function we then derive further equivalent conditions one of them being that p has a stabilizing controller with internal loop this and some others are new even in the scalar valued case we also establish certain related results for example we extend the classical results on coprime factorization and partial feedback measurement feedback stabilization to nonrational transfer functions all our results apply in both discrete and continuous time settings except that in the latter it is not clear whether the controller q can always be chosen so that it is continuous time proper holomorphic and bounded on a right half plane unless e g p z rightarrow0 as operatorname re z rightarrow infty

streamline topologies near a stationary wall of stokes flow in a cavity two dimensional stokes flow in a rectangular single lid driven cavity with the upper lid moving is considered to investigate streamline topologies near a stationary wall by using both of the qualitative theory and an analytic solution we consider the velocity field by expanding in a taylor series at the origin a systematic use of normal form transformations results in a much simplified system of differential equations for the streamlines encapsulating all features of the original system using the simplified system a sequence of possible bifurcations of streamlines is obtained close to a simple linear degeneracy then to extract the same streamline topologies by varying a physical parameter the cavity aspect ratio we used an eigenfunction solution for the streamfunction

a novel velocity vorticity formulation of the navier stokes equations with applications to boundary layer disturbance evolution a novel velocity vorticity formulation of the unsteady three dimensional navier stokes equations is presented the formulation is particularly suitable for simulating the evolution of three dimensional disturbances in boundary layers a key advantage is that there are only three governing equations for three primary dependent variables another advantage is that no wall boundary conditions are needed for the vorticity instead the conditions placed on the velocity are linked to the vorticity field through integral constraints based on the definition of vorticity numerical methods are presented in the context of application to the three dimensional boundary layer over a rotating disc the discretization scheme uses spectral expansions in the wall normal and azimuthal or spanwise directions and compact finite differences in the radial or streamwise direction the scheme is implemented so that the advantages of spectral convergence can be combined with the use of an efficient line iteration solution procedure the linearized form of the new velocity vorticity method is validated for the case of convective instabilities evolving over both rigid and compliant discs

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

a splitting theorem for the space of smooth functions we will show using purely linear functional analytic methods that each exact complex0 f c s0 p0 c s1 p1 c s2 p2 wherepiare matrices of convolution operators in particular linear differential operators with constant coefficients splits fromp1on in the category of topological vector spaces moreover we characterize when these complexes split completely the obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi the result is a consequence of studying splitting of short exact sequences0 f jx qg 0 wheref x gare graded frechet spaces i e frechet spaces equipped with fixed reduced spectra representing them andj qare consistent with the graded structures we characterize splitting of in case i forgare spacesc where is an open subset in rn or ii gis the space of all sequences