brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

accuracy improvement of extremum seeking control in this work we present a modification for the classic and phasor extremum seeking control algorithms in order to improve the accuracy by removing or reducing the convergence error the modulation signals were replaced by a sum of sinusoids in order to remove the equilibrium shift in the controlled variable of the averaged system the convergence error is calculated as a function of the number of sinusoids used in the modulation signal a simulation example is presented to illustrate the improvement

newton based stochastic extremum seeking in this paper we introduce a newton based approach to stochastic extremum seeking and prove local stability of newton based stochastic extremum seeking algorithm in the sense of both almost sure convergence and convergence in probability the convergence of the newton algorithm is proved to be independent of the hessian matrix and can be arbitrarily assigned which is an advantage over the standard gradient based stochastic extremum seeking simulation shows the effectiveness and advantage of the proposed algorithm over gradient based stochastic extremum seeking

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

finite elements and object oriented implementation techniques in computational fluid dynamics abstract a unified flexible and extensible framework for simulating viscous fluid flow using object oriented programming techniques is presented with this framework at hand it is easy to implement and test new solution approaches and to compare available solution methods in a given physical problem several strategies for solving the navier stokes equations are implemented and tested and some test cases are presented an extension of the framework to turbulence modeling and two phase flow is also demonstrated the flexibility and modularity of cfd codes that we advocate in the present paper are considerable easier to achieve using object orientation than the programming techniques that currently dominate the field of cfd

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

application of time iterative schemes to incompressible flow the computation of steady incompressible flows by an euler implicit algorithm is studied using both the incompressible equations and the low mach number compressible equations the incompressible equations are handled by adding an artificial time derivative to the continuity equation this allows both the pressure and velocity to be obtained implicitly in one dimensional problems both systems converge rapidly even at low mach numbers where the eigenvalues are very stiff in two dimensions where approximate factorization is required the presence of stiff eigenvalues is highly detrimental stiffness can be avoided in the incompressible equations by selecting an appropriate pseudo mach number this insures reliable convergence and results in an efficient incompressible flow algorithm in the case of compressible equations the mach number cannot be chosen arbitrarily and the contamination introduced by approximate factorization must be removed a matrix preconditioning factor that accomplishes this is developed and demonstrated with this modification the convergence rate is the same as in the incompressible case and is independent of mach number rapid convergence is observed at mach numbers as low as 6 05

hitchin kobayashi correspondence quivers and vortices a twisted quiver bundle is a set of holomorphic vector bundles over a com plex manifold labelled by the vertices of a quiver linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles labelled by the arrows when the manifold is kahler quiver bundles admit natural gauge theoretic equations which unify many known equations for bundles with extra structure in this paper we prove a hitchin kobayashi correspondence for twisted quiver bundles over a compact kahler manifold relating the existence of solutions to the gauge equations to a stability criterion and con sider its application to a number of situations related to higgs bundles and dimensional reductions of the hermitian einstein equations

discrete singular convolution finite subdomain method for the solution of incompressible viscous flows this paper proposes a discrete singular convolution finite subdomain method dsc fsm for the analysis of incompressible viscous flows in multiply connected complex geometries the dsc algorithm has its foundation in the theory of distributions a block structured grid of fictitious overlapping interfaces is designed to decompose a complex computational geometry into a finite number of subdomains in each subdomain the governing navier stokes equations are discretized by using the dsc algorithm in space and a third order runge kutta scheme in time information exchange between fictitious overlapping zones is realized by using the dsc interpolating algorithm the taylor problem with decaying vortices could be solved to machine precision with an excellent comparison against the exact solution the reliability of the proposed method is tested by simulating the flow in a lid driven cavity the utility of the dsc fsm approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results compare well with the numerical and experimental data available in the literature

pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations a calculation procedure for viscous flows at all mach numbers has been described the scheme has been developed for a nonorthogonal coordinate system in order to handle both incompressible and compressible flows pressure has been selected as a primary dependent variable in preference to density the pressure field is evaluated using the compressible form of the simpler algorithm the accuracy of the proposed scheme has been assessed by comparing the results with experimental data or other numerical results available in the literature

multivariable extremum seeking for pde dynamic systems in this article we address the design and analysis of multivariable newton based extremum seeking es for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations pdes multi input systems with distinct diffusion coefficients in each individual input channel are dealt with the phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation based averaging based estimate of the hessian s inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the diffusion pdes in addition the generalization of the results for different classes of parabolic reaction advection diffusion pdes wave equations and or first order hyperbolic transport dominated pdes is also discussed as for ordinary differential equations case the proposed newton approach removes the dependence of the algorithm s convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable unlike the gradient algorithm a numerical example illustrates the performance of our multivariable es for compensating pde based systems

quantization of donaldson s heat flow over projective manifolds consider e a holomorphic vector bundle over a projective manifold x polarized by an ample line bundle l fix k large enough the holomorphic sections h 0 e otimes l k provide embeddings of x in a grassmanian space we define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of x this flow can be seen as a flow of algebraic type hermitian metrics on e at the quantum limit k rightarrow infty we prove the convergence of the balancing flow towards the donaldson heat flow up to a conformal change as a by product we obtain a numerical scheme to approximate the yang mills flow in that context

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

a discrete splitting finite element method for numerical simulations of incompressible navier stokes flows the presence of the pressure and the convection terms in incompressible navier stokes equations makes their numerical simulation a challenging task the indefinite system as a consequence of the absence of the pressure in continuity equation is ill conditioned this difficulty has been overcome by various splitting techniques but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity whenever introduced we present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions the non linear convection term can be treated by four different approaches and here we present a new linear implicit time scheme these two new techniques are implemented with a finite element method and numerical verifications are made copyright 2005 john wiley sons ltd

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

multivariable newton based extremum seeking we present a newton based extremum seeking algorithm for the multivariable case the design extends the recent newton based extremum seeking algorithms for the scalar case and introduces a dynamic estimator of the inverse of the hessian matrix that removes the difficulty with the possible singularity of a possible direct estimate of the hessian matrix the estimator of the inverse of the hessian has the form of a differential riccati equation we prove local stability of the new algorithm for general nonlinear dynamic systems using averaging and singular perturbations in comparison with the standard gradient based multivariable extremum seeking the proposed algorithm removes the dependence of the convergence rate on the unknown hessian matrix and makes the convergence rate of both the parameter estimates and of the estimates of the hessian inverse user assignable in particular the new algorithm allows all the parameters to converge with the same speed yielding straight trajectories to the extremum even with maps that have highly elongated level sets in contrast to curved steepest descent trajectories of the gradient algorithm simulation results show the advantage of the proposed approach over gradient based extremum seeking by assigning equal desired convergence rates to all the parameters using newton s approach

observer based perturbation extremum seeking control with input constraints for direct contact membrane distillation process abstractan observer based perturbation extremum seeking control is proposed for a direct contact membrane distillation dcmd process the process is described with a dynamic model that is based on a 2d advection diffusion equation model which has pump flow rates as process inputs the objective of the controller is to optimise the trade off between the permeate mass flux and the energy consumption by the pumps inside the process cases of single and multiple control inputs are considered through the use of only the feed pump flow rate or both the feed and the permeate pump flow rates a nonlinear lyapunov based observer is designed to provide an estimation for the temperature distribution all over the designated domain of the dcmd process moreover control inputs are constrained with an anti windup technique to be within feasible and physical ranges performance of the proposed structure is analysed and simulations based on real dcmd process parameters for each control input are provided

semi implicit extension of a godunov type scheme based on low mach number asymptotics i one dimensional flow a single scale multiple space scale asymptotic analysis provides detailed insight into the low mach number limit behavior of solutions of the compressible euler equations we use the asymptotics as a guideline for developing a low mach number extension of an explicit higher order shock capturing scheme this semi implicit scheme involves multiple pressure variables large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step this equation is a pressure correction equation for incompressible flows when the mach number is zero in the low mach number limit the time step is restricted by a courant number based essentially on the maximum flow velocity for moderate and large mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm

extremum seeking under stochastic noise and applications to mobile sensors in this paper the extremum seeking algorithm with sinusoidal perturbations has been extended and modified in two ways a the output of the system is corrupted with measurement noise b the amplitudes of the perturbation signals as well as the gain of the integrator block are time varying and tend to zero at a pre specified rate convergence to the extremal point with probability one has been proved also as a consequence of being able to cope with a stochastic environment it has been shown how the proposed algorithm can be applied to mobile sensors as a tool for achieving the optimal observation positions the proposed algorithm has been illustrated through several simulations

newton like extremum seeking for the control of thermoacoustic instability in practice the convergence rate and stability of perturbation based extremum seeking schemes can be very sensitive to the curvature of the plant map an example of this can be seen in the use of extremum seeking to reduce the amplitude of thermoacoustic oscillations in premixed gas turbine combustors this sensitivity to the plant map curvature arises from the use of a gradient descent adaptation algorithm such extremum seeking schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions resulting in slower optimization than could be achieved for a fixed operating condition this can severely reduce the effectiveness of perturbation based extremum seeking schemes in some applications in this paper a sinusoidally perturbed extremum seeking scheme using a newton like step is developed non local stability results for the scheme are formulated using a lyapunov analysis a local analysis of the scheme is given to investigate the influence of plant dynamics and to show that the local rate of convergence is independent of the plant map curvature the benefit of this plant map curvature independence is then experimentally demonstrated in minimizing the thermoacoustic oscillations in a model premixed combustor

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

a conservative discrete compatibility constraint low mach pressure correction algorithm for time accurate simulations of variable density flows in this paper we develop the discrete compatibility constraint pressure correction algorithm for transient simulations of variable density flows at low mach numbers the constraint for the velocity field is constructed from a combination of the discrete equations of continuity and scalar e g energy transport imposing that the newly predicted state must be compatible in agreement with the equation of state this way mass and scalar conservation are guaranteed and the equation of state is exactly fulfilled at every time step for comparison reasons two other types of well known pressure correction algorithms are also used the first class denoted as continuity constraint pressure correction is based on a constraint for the velocity field that is derived solely from the continuity equation the second class denoted as analytical compatibility constraint pressure correction constructs the constraint from an analytical combination of the material derivative of the equation of state and the continuity and scalar equations the algorithms are tested for three example fluid configurations a single fluid ideal gas a two fluid inert mixture and a two fluid reacting mixture the latter is special in the sense that the equation of state is non linear and not everywhere differentiable the continuity constraint pressure correction algorithm yields unstable solutions if density ratios are high the analytical compatibility constraint pressure correction algorithm yields stable results but the predicted states do not correspond to the equation of state the discrete compatibility constraint pressure correction algorithm performs well on all test cases the simulation results are stable and exactly match the equation of state

twisted holomorphic chains and vortex equations over non compact kahler manifolds abstract in this paper we study twisted holomorphic chains and related gauge equations over non compact kahler manifolds we use the heat flow method to solve the dirichlet boundary problem for the related gauge equations and prove a hitchin kobayashi type correspondence for twisted holomorphic chain over some non compact kahler manifolds

the momentum interpolation method based on the time marching algorithm for all speed flows the time marching approach has clear physical meaning and strict mathematical nature and has been applied in computation of compressible flows widely and extended to many uniform algorithms for all speed flows remedy for its weakness in the problem of checkerboard decoupling of pressure field for incompressible flows is envisaged with the time marching momentum interpolation method mim taken into account in this paper existing preconditioning methods for suppressing decoupling and time marching mim are analyzed for this purpose and algorithms of time marching mim are proposed for steady and unsteady flows and for all speed flows asymptotic analysis shows that the supposed time marching mim has at least a third order accuracy better than the existing time marching coupling methods which only have an accuracy of the same order as the adopted scheme has effects of the time step sizes on the ability of the time marching mim to suppress the checkerboard pressure decoupling are particularly discussed in terms of the dual time stepping approach and it is revealed how the decreased sizes of either the pseudo or physical time step increases the possibility of decoupling and how choi s modification in which the history of the interface velocity is decided by itself instead of the arithmetic average of the velocities on its adjacent nodes removes the unphysical pressure oscillation with small size of the physical time step but leads to divergence with the pseudo time step as well as a remedy for the pseudo time step such methods are recommended as implicit methods and the local time step method with a proposed modification of the time marching mim preventing accuracy loss due to very large time step size numerical experiments support the theoretical analyses and show the validity of the time marching mim proposed

analysis of an asymptotic preserving scheme for the euler poisson system in the quasineutral limit in a previous work p crispel p degond and m h vignal j comput phys 223 2007 pp 208 234 a new numerical discretization of the euler poisson system was proposed this scheme is asymptotic preserving in the quasineutral limit i e when the debye length varepsilon tends to zero which means that it becomes consistent with the limit model when varepsilon to 0 in the present work we show that the stability domain of the present scheme is independent of varepsilon this stability analysis is performed on the fourier transformed with respect to the space variable linearized system we show that the stability property is more robust when a space decentered scheme is used which brings in some numerical dissipation rather than a space centered scheme the linearization is first performed about a zero mean velocity and then about a nonzero mean velocity at the various stages of the analysis our scheme is compared with more classical schemes and its improved stability property is outlined the analysis of a fully discrete in space and time version of the scheme is also given finally some considerations about a model nonlinear problem the burgers poisson problem are also discussed

extremum seeking for static maps with delays in this paper we address the design and analysis of multi variable extremum seeking for static maps subject to arbitrarily long time delays both gradient and newton based methods are considered multi input systems with different time delays in each individual input channel as well as output delays are dealt with the phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation based averaging based estimate of the hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point even in the presence of delays the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the time delay in particular a new backstepping like transformation is introduced to design the predictor for the gradient based extremum seeking scheme with multiple and distinct input delays the proposed newton based extremum seeking approach removes the dependence of the convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable as in the literature free of delays a source seeking example illustrates the performance of the proposed delay compensated extremum seeking schemes

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a distributed simultaneous perturbation approach for large scale dynamic optimization problems we consider distributed optimization problems of large scale dynamic systems where the global cost is the sum of all individual costs of subsystems which are only known to the associated agent to solve this problem a distributed simultaneous perturbation approach d spa is proposed based on simultaneous perturbation techniques as well as consensus theory the proposed method is model free so long as all individual costs can be measured and requires little knowledge on the coupling structure of the problem to be optimized the convergence of the proposed scheme is proved using singular perturbation and averaging theory in particular with proper choice of the parameters under design we show that the proposed scheme is able to converge to the neighborhood of the pareto optimum of the problem so long as the energy of perturbation signals is sufficiently small moreover the proposed approach is applied to a simulated offshore wind farm for energy maximization and a comprehensive comparison with the existing state of the art technique is made to illustrate its effectiveness

extremum seeking for limit cycle minimization in many physical problems equilibrium stabilization is not possible and the controlled system is in a limit cycle if the size of the limit cycle depends on some of the control parameters then a reasonable objective would be to tune this parameter to minimize the size of the limit cycle we propose a method for achieving this this method is an extension of our earlier result kristic and wang 2000 on extremum seeking for equilibria we illustrate the method with a van der pol oscillator example and present analysis for it using averaging and singular perturbations

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

a high resolution pressure based algorithm for fluid flow at all speeds abstract a new collocated finite volume based solution procedure for predicting viscous compressible and incompressible flows is presented the technique is equally applicable in the subsonic transonic and supersonic regimes pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the newly developed algorithm has two new features i the use of the normalized variable and space formulation nvsf methodology to bound the convective fluxes and ii the use of a high resolution scheme in calculating interface density values to enhance the shock capturing property of the algorithm the virtues of the newly developed method are demonstrated by solving a wide range of flows spanning the subsonic transonic and supersonic spectrum results obtained indicate higher accuracy when calculating interface density values using a high resolution scheme

a projection method for low speed flows we propose a decomposition applicable to low speed inviscid flows of all mach numbers less than 1 by using the hodge decomposition we may write the velocity field as the sum of a divergence free vector field and a gradient of a scalar function evolution equations for these parts are presented a numerical procedure based on this decomposition is designed using projection methods for solving the incompressible variables and a backward euler method for solving the potential variables numerical experiments are included to illustrate various aspects of our algorithm

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the second order projection method for the backward facing step flow abstract in an effort to demonstrate the validity of the second order projection method for incompressible and viscous flows inside irregular domains we use the method to study the flow in the two dimensional channel with a backward facing step the emphasis of our study is on the flow separation and reattachment phenomena calculations are performed over a wide range of reynolds numbers including the laminar transitional and turbulent regimes defined by experiments in the laminar regime we find excellent agreement with experiments in terms of the dependence of the reattachment length on the reynolds number and streamwise velocity profiles for reynolds numbers beyond the laminar regime where there are no two dimensional experimental data available for comparison we find characteristic flow structures similar to those observed in three dimensional experiments for such physically unstable flows we are able to show certain numerical convergence in the statistical behavior of the solution comparison with the random vortex method is also made and similar flow behavior is found in both solutions

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

extremum seeking control based on phasor estimation we present an extremum seeking control algorithm based on the estimation of the phasor of the perturbation frequency in the output of the plant the phasor estimator is based on a continuous time k

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

fast extremum seeking on hammerstein plants partial plant knowledge may be used to develop model based extremum seekers however existing results rely on a type of time scale separation which leads to slow optimization relative to the plant dynamics in this work a fast model based extremum seeking scheme is proposed for a hammerstein plant and semi global stability results are provided the structure of a hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time scale than the plant dynamics this leads to fast convergence while maintaining semi global stability

performance of compressible flow codes at low mach numbers the accuracy and the performance of three two dimensional compressible flow codes at freestream mach numbers as low as 0 001 are examined two of the codes employ a finite volume discretization scheme along with a multistage time stepping algorithm to solve the euler equations the two codes differ in their respective use of cell centered and node centered differencing schemes the third code uses an implicit finite difference procedure to solve the unsteady navier stokes equations computational test cases are the inviscid steady flow over a circular cylinder and the impulsively started viscous flow over a cylinder

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

balanced metrics on twisted higgs bundles a twisted higgs bundle on a kahler manifold x is a pair e phi consisting of a holomorphic vector bundle e and a holomorphic bundle morphism phi m otimes e rightarrow e for some holomorphic vector bundle m such objects were first considered by hitchin when x is a curve and m is the tangent bundle of x and also by simpson for higher dimensional base the hitchin kobayashi correspondence for such pairs states that e phi is polystable if and only if e admits a hermitian metric solving the hitchin equation this correspondence is a powerful tool to decide whether there exists a solution of the equation but it provides little information as to the actual solution in this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and balanced metrics that give approximate solutions to the hitchin equation motivation for this study comes from work of donagi wijnholt jhep 05 068 2013 concerning balanced metrics for the vafa witten equations

invariant connections and vortices we study the vortex equations on a line bundle over a compact kahler manifold these are a generalization of the classical vortex equations over 2 we first prove an invariant version of the theorem of donaldson uhlenbeck and yau relating the existence of a hermitian yang mills metric on a holomorphic bundle to the stability of such a bundle we then show that the vortex equations are a dimensional reduction of the hermitian yang mills equation using this fact and the theorem above we give a new existence proof for the vortex equations and describe the moduli space of solutions

a perturbation based proportional integral extremum seeking control approach this paper proposes a proportional integral extremum seeking control technique the technique is a generalization of the standard perturbation based techniques which provides fast transient performance of the closed loop system to the optimum equilibrium of a measured cost function under the assumption that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown it is shown that for a class of minimum phase nonlinear system with a cost function of relative degree one the closed loop esc can practically stabilize the unknown equilibrium optimum the stabilization result provides an alternative approach to output feedback control of nonlinear systems the effectiveness of the proposed approach is demonstrated using simulation examples

an all speed roe type scheme and its asymptotic analysis of low mach number behaviour a new scheme all speed roe scheme was proposed for all speed flows compared with traditional preconditioned roe scheme all speed roe scheme changes non linear eigenvalues in the numerical dissipation terms of roe type schemes with an asymptotic analysis the low mach number behaviour of the scheme is studied theoretically in two ways in one way all speed roe scheme is regarded as finite magnification of low speed roe scheme in the low mach number limit in the other way a general form of all speed roe scheme is analyzed both ways demonstrate that all speed roe scheme has the same low mach number behaviour as the original governing equation in the continuous case which includes three important features pressure variation scales with the square of the mach number the zero order velocity field is subject to a divergence constraint and the second order pressure satisfies a poisson type equation in the case of constant entropy the analysis also leads to an unexpected conclusion that the velocity filed computed by traditional preconditioned roe scheme does not satisfy the divergence constraint as the mach number vanishes moreover the analysis explains the reason of checkerboard decoupling and shows that momentum interpolation method provides a similar mechanism as traditional preconditioned roe scheme inherently possesses to suppress checkerboard decoupling in the end general rulers for modifying non linear eigenvalues are obtained finally several numerical experiments are provided to support the theoretical analysis all speed roe scheme has a sound foundation and is expected to be widely studied and applied to all speed flow calculations

mach uniformity through the coupled pressure and temperature correction algorithm we present a new type of algorithm the coupled pressure and temperature correction algorithm it is situated in between the fully coupled and the fully segregated approach and is constructed such that mach uniform accuracy and efficiency are obtained the essential idea is the separation of the convective and the acoustic thermodynamic phenomena a convective predictor is followed by an acoustic thermodynamic corrector for a general case the corrector consists of a coupled solution of the energy and the continuity equations for both pressure and temperature corrections for the special case of an adiabatic perfect gas flow the algorithm reduces to a fully segregated method with a pressure correction equation based on the energy equation various test cases are considered which confirm that mach uniformity is obtained

numerical approximations of pressureless and isothermal gas dynamics we study several schemes of first or second order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations the pressureless gas system is weakly hyperbolic giving rise to the formation of density concentrations known as delta shocks for the isothermal gas system the infinite speed of expansion into vacuum leads to zero timestep in the godunov method based on exact riemann solver the schemes we consider are able to bypass these difficulties they are proved to satisfy positiveness of density and discrete entropy inequalities to capture the delta shocks and to treat data with vacuum

the construction of an optimal weakly divergence free macroelement the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and reduces significantly the dimension of the system to be solved at each time step for the standard 8 node velocity and 4 node pressure dffem an optimal basis for the weakly divergence free subspace is constructed such that each basis function has non zero support on at most nine contiguous elements given this basis weakly divergence free macroelements are constructed

a superlinearly convergent mach uniform finite volume method for the euler equations on staggered unstructured grids a mach uniform finite volume scheme for solving the unsteady euler equations on staggered unstructured triangular grids that uses linear reconstruction is described the scheme is applied to three benchmark problems and is found to be considerably more accurate than a similar scheme based on piecewise constant reconstruction

a framework for a class of hybrid extremum seeking controllers with dynamic inclusions abstract this paper presents a prescriptive framework for the design and analysis of a class of deterministic extremum seeking controllers escs based on hybrid dynamic inclusions this type of escs combines continuous and discrete time dynamics during the seeking process and its evolution in time is characterized by differential and difference inclusions rather than standard difference and differential equations examples of hybrid escs include but are not limited to purely continuous esc with optimizers described by set valued mappings esc with arbitrarily fast and slow switching modes esc with weakly jumping parameters as well as distributed escs for multi agent systems with time varying graphs since solutions of systems described by set valued mappings are usually not unique we do not insist on this property but rather we characterize the behavior of all possible solutions generated by the closed loop system making use of recent results in singular perturbations averaging and limit sets of sets for hybrid dynamical systems we establish a practical asymptotic stability result some examples of common classes of seeking dynamics described by hybrid systems are presented as well as some numerical simulations illustrating their application in standard maximization problems and nash seeking problems in game theoretical scenarios

upwind finite volume method with a triangular mesh for conservation laws a new upwind scheme has been developed and analyzed for a finite volume solution of the conservation laws on triangular meshes the scheme is an upwind second order extrapolation with simple local limiters and it is weakly second order accurate and satisfies maximum principles in one dimension the scheme reduces to a fully upwind second order scheme with a simple local limiter preliminary numerical results demonstrating the performance of the scheme on a variety of initial boundary value problems are presented the order of convergence of the scheme is found to vary from 1 6 to 1 9 in l1

stability of approximate projection methods on cell centered grids projection methods are a popular class of methods for solving the incompressible navier stokes equations if a cell centered grid is chosen in order to use high resolution methods for the advection terms performing the projection exactly is problematic an attractive alternative is to use an approximate projection in which the velocity is required to be only approximately discretely divergence free the stability of the cell centered approximate projection is highly sensitive to the method used to update the pressure and compute the pressure gradient this is demonstrated by analyzing a model problem and conducting numerical simulations of the navier stokes equations

incompressibility without tears how to avoid restrictions of mixed formulation the severe restrictions of the babuska brezzi stability criteria permit only a limited number of interpolations for velocities displacements and pressure to be used for incompressible behaviour these restrictions preclude the use of many useful elements we show in this paper how such difficulties can be side stepped by seeking the steady state solution through the use of various time marching schemes this permits a simple iterative approach to incompressible or nearly incompressible problems of fluid mechanics or solid mechanics and provides a stimulus for the use of such procedures in metal forming flow etc

vortex type equations and canonical metrics we introduce a notion of gieseker stability for a filtered holomorphic vector bundle mathcal f over a projective manifold we relate it to an analytic condition in terms of hermitian metrics on mathcal f coming from a construction of the geometric invariant theory g i t we prove that if there is a hermite einstein metric h he on mathcal f then there exists a sequence of such balanced metrics that converges and its limit is h he as a corollary we obtain an approximation theorem for quiver vortex equations and other classical equations

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

maximizing map sensitivity and higher derivatives via extremum seeking we present a generalization to the scalar newton based extremum seeking algorithm which through perturbation induced measurements of an unknown steady state input to output map maximizes the map s higher derivatives as with other extremum seeking es problems the scheme relies on derivative estimators and learning dynamics operating in different time scales we provide analysis for the typical deterministic sinusoidal perturbation of the optimal parameter estimate also we alternatively include a stochastic method where the map is perturbed via the sinusoid of brownian motion about the boundary of a circle by properly demodulating the map output corresponding to the manner in which it is perturbed the es algorithm maximizes the n rm th derivative only through measurements of the map the newton based es approach removes the dependence of the convergence rate on the unknown hessian of the higher derivative an effort to improve performance over standard gradient based es our design stems from the existing multivariable newton based es algorithm where a differential riccati equation estimates the inverse hessian of the function to be maximized algebraically computing a direct estimate of the inverse hessian is susceptible to singularity whereas employing the riccati filter removes that potential we prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and compare the newton based method against the gradient based method we also extend this abstraction to multiplayer noncooperative games but limit our attention to a two player game for notational simplicity and length considerations

dimensional reduction and quiver bundles the so called hitchin kobayashi correspondence proved by donaldson uhlenbeck and yau establishes that an indecomposable holomorphic vector bundle over a compact kahler manifold admits a hermitian einstein metric if and only if the bundle satisfies the mumford takemoto stability condition in this paper we consider a variant of this correspondence for g equivariant vector bundles on the product of a compact kahler manifold x by a flag manifold g p where g is a complex semisimple lie group and p is a parabolic subgroup the modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle the study of invariant solutions to the modified hermitian einstein equation over xg p leads via dimensional reduction techniques to gauge theoretic equations on x these are equations for hermitian metrics on a set of holomorphic bundles on x linked by morphisms defin ing what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup p similarly the corresponding stability condition for the invariant filtration over xg p gives rise to a stability condition for the quiver bundle on x and hence to a hitchin kobayashi correspondence in the simplest case when the flag manifold is the complex projective line one recovers the theory of vortices stable triples and stable chains as studied by bradlow the authors and others

extremum seeking control for discrete time systems we present an extremum seeking control algorithm for discrete time systems applied to a class of plants that are represented as a series combination of a linear input dynamics a static nonlinearity with an extremum and a linear output dynamics by using the two time scale averaging theory we derive a mild sufficient condition under which the plant output exponentially converges to an o spl alpha sup 2 neighborhood of the extremum value where spl alpha is the magnitude of modulation signal the sufficient condition is related to positive realness of linear parts of the plant but only at the modulation frequency the algorithm is illustrated with a brief simulation study