high order difference schemes for unsteady one dimensional diffusion convection problems abstract for unsteady 1d diffusion convection problems this paper develops an extensive analysis of two level three point finite difference schemes of order 2 in time and 4 in space this general class of fds includes several schemes independently proposed by different authors one main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems i e when the cell reynolds number b h 2 is greater than unity the stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments we outline some key points i the great difficulty to obtain accurate numerical results for large values of ii the possibility of virtually optimum schemes is essentially theoretical and requires in practice careful experiments iii for strongly convective problems some second order explicit schemes are almost as efficient and less costly than implicit fourth order schemes

stability and convergence of the peaceman rachford adi method for initial boundary value problems etude des proprietes de stabilite et de convergence de la methode des directions alternees implicite losrsqu elle est appliquee aux problemes aux valeurs initiales dans un espace bidimensionnel

quantum gate generation by t sampling stabilisation this paper considers right invariant and controllable driftless quantum systems with state x t evolving on the unitary group u n and m inputs u u1 um the t sampling stabilisation problem is introduced and solved given any initial condition x0 and any goal state find a control law u u x t such that for the closed loop system the purpose is to generate arbitrary quantum gates corresponding to this is achieved by the tracking of t periodic reference trajectories of the quantum system that pass by using the framework of coron s return method the t periodic reference trajectories are generated by applying controls that are a sum of a finite number m of harmonics of sin 2 t t whose amplitudes are parameterised by a vector a the main result establishes that for m big enough x jt exponentially converges towards for almost all fixed a with explicit and completely constructive control laws this paper also establishes a stochastic version of this deterministic control law the key idea

multigrid solution of the euler equations using implicit schemes on utilise une methode multigrille du type factorisation approchee couplee a un schema de direction alternee implicite pour resoudre les equations d euler pour l ecoulement transsonique sur un profil aerodynamique on obtient une convergence tres rapide

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

application of the parallel diagonal dominant algorithm for the incompressible navier stokes equations abstract the accuracy and the applicability of the parallel diagonal dominant pdd algorithm are explored for highly scalable computation of the incompressible navier stokes equations which are integrated using a fully implicit fractional step method in parallel computational environments the pdd algorithm is known to be applicable only for an evenly diagonal dominant matrix in the present study however it is shown mathematically that the pdd algorithm is utilizable even for non diagonal dominant matrices derived from discretization of incompressible momentum equations the order of accuracy and the error characteristics are investigated in detail in terms of the courant friedrichs lewy cfl number and the grid spacing by conducting simulations of decaying vortices in both two and three dimensions flow in a lid driven cavity and flow over a circular cylinder in order to reduce communication cost which is one of bottlenecks in parallel computation an aggregative data communication method is combined with the pdd algorithm parallel performance of the present pdd based method is investigated by measuring the speedup efficiency overhead and serial fraction

stability and error estimates of a new high order compact adi method for the unsteady 3d convection diffusion equation abstract in this paper a new high order compact adi method for the unsteady convection diffusion equation in three dimension 3d is considered collecting the truncation error of the finite difference operator by the recursion method we derive a new high order compact finite difference scheme for the unsteady 1d convection diffusion equation firstly then based on the adi method applying a correction technique to reduce the error of the splitting term a high order compact adi method for the unsteady 3d convection diffusion equation is proposed in which we solve a series of 1d problems with strictly diagonal dominant tri diagonal structures instead of the high dimensional ones the scheme is proved to be unconditional stable moreover the regularity and error estimate of the numerical solution are derived finally some numerical examples are performed which confirm the theoretical prediction and show that much better computational accuracy results can be got by applying the new scheme

higher order adi method with completed richardson extrapolation for solving unsteady convection diffusion equations a higher order alternating direction implicit adi method with completed richardson extrapolation is proposed for solving unsteady two dimensional convection diffusion equations the method has sixth order accuracy in space and fourth order accuracy in time completed richardson extrapolation is used to improve the accuracy of the solution obtained from the high order adi method in spatial and temporal domains simultaneously stability analysis is given to show the influence of richardson extrapolation on the unconditionally stable adi method numerical experiments are conducted to test its high accuracy and to compare it with karaa zhang s high order adi method

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

locally distinguishing inputs for systems on lie groups and parallelism this paper is concerned with invariant systems on lie groups a sufficient condition for a piecewise constant input to distinguish between any two distinct initial states contained in some open neighbourhood of the state space is given this condition involves the parallel translation of vectors along curves thus relating observability to parallelism

suitability of upwind biased finite difference schemes for large eddy simulation of turbulent flows we have simulated flow past a circular cylinder at a reynolds number of 3 9 x 10 3 using a solver that employs an energy conservative second order central difference scheme for spatial discretization detailed comparisons of turbulence statistics and energy spectra in the downstream wake region 7 0 x d 10 0 have been made with the results of beaudan and moin and with experiments to assess the impact of numerical diffusion on the flowfield based on these comparisons conclusions are drawn on the suitability of higher order upwind schemes for les in complex geometries

control aspects of holonomic quantum computation a unifying framework for the control of quantum systems with non abelian holonomy is presented it is shown that from a control theoretic point of view holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle an extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm

analysis of material interface discontinuities and superconvergent fluxes in finite difference theory abstract an analysis of material interface discontinuities is developed and applied in finite difference theory to determine mathematically rigorous averaging techniques for material properties this result is compared with other averaging techniques particularly harmonic averaging which is often applied in practice we also develop a class of formulas of high accuracy for post processing the difference formula to compute derivatives fluxes stresses and conduct supporting numerical studies

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

the critical value for some non attractive long range nearest particle systems we consider the nearest particle system which gives birth rate to each vacant interval concentrated on the interval s midpoint s we prove that a critical value for exists and equals one the proof extends to a large class of nearest particle systems this paper solves a problem suggested by liggett 1985

reduction of sufficient conditions for optimal control problems with subgroup symmetry in this paper we apply symmetry reduction techniques from geometric mechanics to sufficient conditions for local optimality in optimal control problems after reinterpreting some previous results for left invariant problems on lie groups we focus on optimal control problems with subgroup symmetry for these problems the necessary conditions for optimality can be simplified by exploiting symmetries so as to reduce the number of variables needed to describe trajectories of the system we show that sufficient conditions for optimality based on the non existence of conjugate points can be simplified in an analogous way to the necessary conditions we demonstrate these simplifications by analyzing an optimal control problem that models a spinning top in a gravitational field and we give particular attention to the example of an axisymmetric sleeping top the results we derive in this paper allow us to determine which trajectories of a sleeping top are locally optimal solutions of the optimal control problem which is a new result that has not appeared in previous literature

a numerical method for large eddy simulation in complex geometries we discuss the development of a numerical algorithm and solver capable of performing large eddy simulation in the very complex geometries often encountered in industrial applications the algorithm is developed for unstructured hybrid grids is non dissipative yet robust at high reynolds numbers on highly skewed grids simulation results for a variety of flows are presented

a high order exponential scheme for solving 1d unsteady convection diffusion equations in this paper a high order exponential hoe scheme is developed for the solution of the unsteady one dimensional convection diffusion equation the present scheme uses the fourth order compact exponential difference formula for the spatial discretization and the 2 2 pade approximation for the temporal discretization the proposed scheme achieves fourth order accuracy in temporal and spatial variables and is unconditionally stable numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature the results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems

spectral resolutioned exponential compact higher order scheme srechos for convection diffusion equations a spectral resolutioned exponential compact higher order scheme srechos is developed to solve stationary convection diffusion type of differential equations with constant convection and diffusion coefficients the scheme is o h6 for one dimensional problems and produces a tri diagonal system of equations that can be solved efficiently using thomas algorithm for two dimensional problems the scheme produces an o h4 k4 accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit adi procedure the efficiency of the developed scheme is measured using wave number analysis the analysis shows that the srechos has a much better spectral resolution than any of the existing higher order schemes numerical examples are solved and better performance of the srechos in terms of accuracy over the existing schemes in the literature is demonstrated

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

analysis of the order parameter for uniform nearest particle system the uniform nearest particle system unps is studied which is a continuoustime markov process with state space left 0 1 right z 1 the rigorous upper bound 1 for the order parameter 2 is given by the correlation identity and the fkg inequality then an improvement of this bound is shown in a similar fashion c 1 log 1 for 1 recently mountford proved that the critical value c 1 combining his result and our improved bound implies that if the critical exponent exists it is strictly greater than the mean field value 1 in the weak sense

a new time space domain high order finite difference method for the acoustic wave equation a new unified methodology was proposed in finkelstein and kastner 2007 39 to derive spatial finite difference fd coefficients in the joint time space domain to reduce numerical dispersion the key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies we develop this new time space domain fd method further for 1d 2d and 3d acoustic wave modeling using a plane wave theory and the taylor series expansion new spatial fd coefficients are frequency independent though they lead to a frequency dependent numerical solution we prove that the modeling accuracy is 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation however under the same discretization the new 1d method can reach 2m th order accuracy and is always stable the 2d method can reach 2m th order accuracy along eight directions and has better stability similarly the 3d method can reach 2m th order accuracy along 48 directions and also has better stability than the conventional fd method the advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models in addition we study the influence of the fd stencil length on numerical modeling for 1d inhomogeneous media and derive an optimal fd stencil length required to balance the accuracy and efficiency of modeling a new time space domain high order staggered grid fd method for the 1d acoustic wave equation with variable densities is also developed which has similar advantages demonstrated by dispersion analysis stability analysis and modeling experiments the methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering

steering for a class of dynamic nonholonomic systems we derive control algorithms for a class of dynamic nonholonomic steering problems characterized as mechanical systems with nonholonomic constraints and symmetries we use a simplified geometric framework in developing two types of control algorithms for such systems the first targets local configuration controllability using a perturbation approach to establish results similar to steering using sinusoids the second method builds on this approach by utilizing kinematic steering algorithms for mobile robots in generating more nonlocal motion plans for this class of systems

a robust higher order compact scheme for solving general second order partial differential equation with derivative source terms on nonuniform curvilinear meshes abstract a fourth order compact finite difference scheme is proposed for solving general second order steady partial differential equation pde in two dimension 2d on geometries having nonuniform curvilinear grids in this work the main efforts are focused not only on nonorthogonal curvilinear grids but also on the presence of mixed derivative term and nonhomogeneous derivative source terms in the governing equation this is in turn suitable for solving fluid flow and heat transfer problems governed by navier stokes n s equations on geometries having nonuniform and nonorthogonal curvilinear grids the newly proposed scheme has been applied to solve general second order partial differential equation having analytical solution and some pertinent fluid flow problems namely viscous flows in a lid driven cavity such as trapezoidal cavity using nonorthogonal grid square cavity using distorted grid complicated enclosures using curvilinear grid and mixed convection flow in a bottom wavy wall cavity it is seen to efficiently capture steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions detailed comparison data produced by the proposed scheme for all the test cases are provided and compared with existing analytical as well as established numerical results available in the literature excellent comparison is obtained in all the cases

explicit wei norman formulae for matrix lie groups via putzer s method abstract the wei norman formula locally relates the magnus solution of a system of linear time varying odes with the solution expressed in terms of products of exponentials by means of a set of nonlinear differential equations in the parameters of the two types of solutions a closed form expression of such formula is proposed based on the use of putzer s method

configuration controllability of mechanical systems underactuated by one control we investigate local configuration controllability for mechanical control systems within the affine connection formalism we rely on previous results on controllability and series expansions for the evolution of mechanical systems starting from rest extending the work by lewis for the single input case we are able to characterize local configuration controllability for systems with n degrees of freedom and n 1 input forces

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

optimal control of a co rotating vortex pair averaging and impulsive control abstract the problem of controlling the position of a pair of point vortices using a strain field or a field of a single source sink is considered the problem dimension is reduced by averaging over the fast rotation of vortices around the center of vorticity control is assumed to be function of rotation phase only for the case of the strain field actuation optimal control problem is posed in three different settings control input whose integral over a cycle of vortex rotation is bounded bounded control input and mixed problem where both integral over a cycle and control magnitude are bounded it is shown that optimal solutions in integral setting are impulsive they consist of dirac delta functions applied at optimal phases during the cycle of vortex rotation if a bound on the control magnitude is added to the constraints the mixed problem a solution is obtained for which control is at the maximum amplitude except possibly for intervals during which it is zero these results are obtained by direct optimization for the case of a source sink field we pose the optimal control problem of the averaged equations through the use of the pontryagin maximum principle an obtained solution is a set of dirac delta pulses at optimal phases of the vortex rotation cycle a discussion of these results in the context of vortex merger control is given impulsive control arises here naturally as an optimal control design once averaging method is used

analysis of stability and accuracy of finite difference schemes on a skewed mesh numerical stability and accuracy of finite difference schemes on a skewed non uniform mesh are investigated to provide guidelines for mesh design and for devising appropriate solution methods when mesh skewness is unavoidable in the current analysis a linear advection diffusion equation in a cartesian coordinate system is transformed into a curvilinear one corresponding to a skewed mesh a finite difference approximation of the transformed equation leads to the leading order error terms which are responsible for time step restrictions and numerical instability a truncation error analysis of central difference approximation reveals the effects of mesh non uniformity and skewness on the solution quality in addition a modified wavenumber analysis is performed for the central and upwind difference schemes coupled with time integration methods to examine the effects of mesh skewness angle and flow direction relative to the mesh angle on the numerical stability in general severe mesh skewness leads to restrictions on the allowable cell peclet number and time step size and also increases phase and amplitude errors

on quantum state observability and measurement we consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable the system is assumed to be a finite dimensional quantum control system for which we can influence the dynamics by generating all the unitary evolutions in a lie group we investigate to what extent by an appropriate sequence of evolutions and measurements we can obtain information on the initial state of the system we present a system theoretic viewpoint of this problem in that we study the observability of the system in this context we characterize the equivalence classes of indistinguishable states and propose algorithms for state identification

explicit design of time varying stabilizing control laws for a class of controllable systems without drift abstract this paper gives a systematic way to design time variant feedback control laws for a class of controllable nonlinear systems this class contains a lot of systems which cannot be stabilized via a time invariant feedback control law the interest of this work lies in the design method since a general existence results is already available the techniques employed here are basic they mainly involve classical lyapunov analysis

integral control on lie groups in this paper we extend the popular integral control technique to systems evolving on lie groups more explicitly we provide an alternative definition of integral action for proportional derivative controlled systems whose configuration evolves on a nonlinear space where configuration errors cannot be simply added up to compute a definite integral we then prove that the proposed integral control allows to cancel the drift induced by a constant bias in both first order velocity and second order torque control inputs for fully actuated systems evolving on abstract lie groups we illustrate the approach by 3 dimensional motion control applications

iterated splitting method of high order for time dependent partial differential equations nonlinear chebyshev iteration is applied for solving the implicit relations which arise when an implicit linear multistep method is used in order to integrate semi discrete initial value problems for partial differential equations the approximate inverse occurring in the defect correction process is obtained by employing splitting methods in order to accelerate convergence the chebyshev iteration process is tuned in such a way that the lower frequencies in the iteration error are strongly damped without using a large number of iterations for moderate accuracies this method is already markedly more efficient then conventional adi methods

control of flow induced noise behind a circular cylinder using splitter plates laminar vortex sheddings behind a circular cylinder with and without splitter plates attached to the cylinder base at the reynolds numbers of 100 and 160 are simulated by solving the unsteady two dimensional incompressible navier stokes equations the strouhal number lift and drag rapidly change with the length of the splitter plate acoustic source functions are obtained from the computed near field velocity and pressure using the curle s solution of the lighthill acoustic analogy in the case of no splitter plate the volume quadrupole noise is small at a low mach number compared with the surface dipole noise from the cylinder when a splitter plate is attached to the cylinder there are significant modifications of the dipole and quadrupole sources changes in the noise sources at re 100 are very different from those at re 160 and their differences are closely related to the secondary vortex generated at the tip of the splitter plate scattering effects at the edge of the splitter plate are also considered

optimal kinematic control of an autonomous underwater vehicle this note investigates the motion control of an autonomous underwater vehicle auv the auv is modeled as a nonholonomic system as any lateral motion of a conventional slender auv is quickly damped out the problem is formulated as an optimal kinematic control problem on the euclidean group of motions se 3 where the cost function to be minimized is equal to the integral of a quadratic function of the velocity components an application of the maximum principle to this optimal control problem yields the appropriate hamiltonian and the corresponding vector fields give the necessary conditions for optimality for a special case of the cost function the necessary conditions for optimality can be characterized more easily and we proceed to investigate its solutions finally it is shown that a particular set of optimal motions trace helical paths throughout this note we highlight a particular case where the quadratic cost function is weighted in such a way that it equates to the lagrangian kinetic energy of the auv for this case the regular extremal curves are constrained to equate to the auv s components of momentum and the resulting vector fields are the d alembert lagrange equations in hamiltonian form

brief series expansions for analytic systems linear in control this paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields we present a series expansion that can be computed via explicit recursive expressions and we derive sufficient conditions for uniform convergence over the finite and infinite time horizon furthermore we present a simplified series and convergence analysis for the setting of second order polynomial vector fields the treatment only relies on elementary notions on analytic functions number theory and operator norms

on local observability for invariant systems on lie groups and coset spaces the purpose of this short paper is to provide a proof of a local observability criterion for invariant systems on lie groups and coset spaces the criterion was originally presented with an ad hoc proof based on lie theoretic arguments the alternative proof presented here shows that the criterion essentially comes from the classical observability results involving the observability codistribution and its properties in this way it unifies the two approaches and extends the application field of the classical results

optimal under actuated kinematic motion planning on the ϵ group abstract a global motion planning method is described based on the solution of minimum energy type curves on the frame bundle of connected surfaces of arbitrary constant cross sectional curvature applying the geometric framing of pontryagin s principle gives rise to necessary conditions for optimality in the form of a boundary value problem this arbitrary dimensional boundary value problem is solved using a numerical shooting method derived from a general lax pair solution the paper then specializes to the 3 dimensional case where the lax pair equations are integrable a semi analytic method for matching the boundary conditions is proposed by using the analytic form of the extremal solutions and a closed form solution for the exponential map this semi analytical approach has the advantage that an analytic description of the control accelerations can be derived and enables actuator constraints to be incorporated via time reparametrization the method is applied to two examples in space mechanics the attitude control of a spacecraft with two reaction wheels and the spacecraft docking problem

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

numerical solution of unsteady advection dispersion equation arising in contaminant transport through porous media using neural networks a soft computing approach based on artificial neural network ann and optimization is presented for the numerical solution of the unsteady one dimensional advection dispersion equation ade arising in contaminant transport through porous media a length factor ann method based on automatic satisfaction of arbitrary boundary conditions bcs was chosen for the numerical solution of ade the strength of ann is exploited to construct a trial approximate solution tas for ade in a way that it satisfies the initial or bcs exactly an unsupervised error is constructed in approximating the solution of ade which is minimized by training ann using gradient descent algorithm gda two challenging test problems of ade are considered in this paper in which the first problem has steep boundary layers near x 1 and many numerical methods create non physical oscillation near steep boundaries also for the second problem many numerical schemes suffer from computational noise and instability issues the proposed method is advantageous as it does not require temporal discretization for the solution of the ades as well as it does not suffer from numerical instability the reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods the results show that the present method removes the difficulties arising in the solution of the ades and provides solution with good accuracy

a multi block adi finite volume method for incompressible navier stokes equations in complex geometries an efficient second order accurate finite volume method is developed for a solution of the incompressible navier stokes equations on complex multi block structured curvilinear grids unlike in the finite volume or finite difference based alternating direction implicit adi methods where factorization of the coordinate transformed governing equations is performed along generalized coordinate directions in the proposed method the discretized cartesian form navier stokes equations are factored along curvilinear grid lines the new adi finite volume method is also extended for simulations on multi block structured curvilinear grids with which complex geometries can be efficiently resolved the numerical method is first developed for an unsteady convection diffusion equation then is extended for the incompressible navier stokes equations the order of accuracy and stability characteristics of the present method are analyzed in simulations of an unsteady convection diffusion problem decaying vortices flow in a lid driven cavity flow over a circular cylinder and turbulent flow through a planar channel numerical solutions predicted by the proposed adi finite volume method are found to be in good agreement with experimental and other numerical data while the solutions are obtained at much lower computational cost than those required by other iterative methods without factorization for a simulation on a grid with o 10 5 cells the computational time required by the present adi based method for a solution of momentum equations is found to be less than 20 of that required by a method employing a biconjugate gradient stabilized scheme

extrapolation algorithm of compact adi approximation for two dimensional parabolic equation abstract in this paper we propose a compact alternating direction implicit adi scheme for solving the two dimensional parabolic equation the maximum norm convergence is proven and the convergence rate is second order in time and fourth order in space we develop a richardson extrapolation algorithm to increase the accuracy to sixth order both in time and space numerical experiments show the effectiveness of the method

a comparative study of adi splitting methods for parabolic equations in two space dimensions abstract the main purpose of the paper is a numerical comparison of three integration methods for semi discrete parabolic partial differential equations in two space variables linear as well as nonlinear equations are considered the integration methods are the well known adi method of peaceman and rachford a global extrapolation scheme of the classical adi method to order four and a fourth order four step adi splitting method

an efficient fourth order low dispersive finite difference scheme for a 2 d acoustic wave equation in this paper we propose an efficient fourth order compact finite difference scheme with low numerical dispersion to solve the two dimensional acoustic wave equation combined with the alternating direction implicit adi technique and pade approximation the standard second order finite difference scheme can be improved to fourth order and solved as a sequence of one dimensional problems with high computational efficiency however such compact higher order methods suffer from high numerical dispersion to suppress numerical dispersion the compact and non compact stages are interlinked to produce a hybrid scheme in which the compact stage is based on pade approximation in both y and temporal dimensions while the non compact stage is based on pade approximation in y dimension only stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the courant friedrichs lewy cfl condition the dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact adi scheme and the higher order locally one dimensional lod scheme three numerical examples are solved to demonstrate the accuracy and efficiency of the new method

symmetry reduction in optimal control of multiagent systems on lie groups we study the reduction of degrees of freedom for the equations that determine necessary optimality conditions for extrema in an optimal control problem for a multiagent system by exploiting the physical symmetries of agents where the kinematics of each agent is given by a left invariant control system reduced optimality conditions are obtained using techniques from variational calculus and lagrangian mechanics a hamiltonian formalism is also studied where the problem is explored through an application of pontryagin s maximum principle for left invariant systems and the optimality conditions are obtained as integral curves of a reduced hamiltonian vector field we apply the results to an energy minimum control problem for multiple unicycles

series expansions for the evolution of mechanical control systems this paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time varying external force mechanical systems are presented as second order systems on a configuration manifold via the notion of affine connections the series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of constant formula a convergence analysis is obtained using some analytic functions and combinatorial analysis results this expansion provides a rigorous means of analyzing locomotion gaits in robotics and lays the foundation for the design of motion control algorithms for a large class of underactuated mechanical systems

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

observers for invariant systems on lie groups with biased input measurements and homogeneous outputs this paper provides a new observer design methodology for invariant systems whose state evolves on a lie group with outputs in a collection of related homogeneous spaces and where the measurement of system input is corrupted by an unknown constant bias the key contribution of the paper is to study the combined state and input bias estimation problem in the general setting of lie groups a question for which only case studies of specific lie groups are currently available we show that any candidate observer with the same state space dimension as the observed system results in non autonomous error dynamics except in the trivial case where the lie group is abelian this precludes the application of the standard non linear observer design methodologies available in the literature and leads us to propose a new design methodology based on employing invariant cost functions and general gain mappings we provide a rigorous and general stability analysis for the case where the underlying lie group allows a faithful matrix representation we demonstrate our theory in the example of rigid body pose estimation and show that the proposed approach unifies two competing pose observers published in prior literature

observability and observers for nonlinear systems this paper deals with observability and observers for general nonlinear systems in the non control affine case we characterize systems that are observable independently of the inputs an exponential observer for these systems is also exhibited

computational methodology for large eddy simulation of tip clearance flows a large eddy simulation based flow solver that combines an immersed boundary technique with a curvilinear structured grid has been developed to study the temporal and spatial dynamics of an incompressible rotor tipclearance flow the overall objective of these simulations is to determine the underlying mechanisms for lowpressure fluctuations downstream of the rotor near the end wall salient features of the numerical methodology including the mesh topology the immersed boundary method the treatment of numerical instability for nondissipative schemes on highly skewed meshes and the parallelization of the code for shared memory computer platforms are discussed the computational approach is shown to be capable of capturing the evolution of the highly complicated flowfield characterized by the interaction of distinct blade associated vortical structures with the turbulent end wall boundary layer simulation results are compared with experiments and qualitative as well as quantitative agreement is observed

a global steering method for nonholonomic systems abstract in this paper we present an iterative steering algorithm for nonholonomic systems also called driftless control affine systems and we prove its global convergence under the sole assumption that the lie algebraic rank condition larc holds true everywhere that algorithm is an extension of the one introduced in jean et al 2005 21 for regular systems the first novelty here consists in the explicit algebraic construction starting from the original control system of a lifted control system which is regular the second contribution of the paper is an exact motion planning method for nilpotent systems which makes use of sinusoidal control laws and which is a generalization of the algorithm described in murray and sastry 1993 29 for chained form systems

singularities of optimal control problems on some 6 d lie groups this paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3 d space a lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular riemannian manifolds specifically the 3 d space forms euclidean space e3 the sphere s3 and the hyperboloid h3 the corresponding frame bundles are equal to the euclidean group of motions se 3 the rotation group so 4 and the lorentz group so 1 3 the maximum principle of optimal control shifts the emphasis for these systems to the associated hamiltonian formalism for an integrable case the extremal curves are explicitly expressed in terms of elliptic functions in this paper a study at the singularities of the extremal curves are given which correspond to critical points of these elliptic functions the extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points it is then shown that the projections of the extremals onto the base space called elastica at these singular points are curves of constant curvature and torsion which in turn implies that the oriented vehicles trace helices

observability of systems on lie groups and coset spaces the purpose of this paper is to study the observability of a class of systems for which the state space is a lie group and the output space is a cosec space the study of this type of system was initiated by brockett siam j control 10 1972 pp 265 284 in this paper brockett s observability results are generalized and necessary and sufficient conditions for observability are obtained effective algorithms are established to verify such conditions finally as an application some disturbance decoupling problems are considered

motion control of drift free left invariant systems on lie groups in this paper we address the constructive controllability problem for drift free left invariant systems on finite dimensional lie groups with fewer controls than state dimension we consider small spl epsiv amplitude low frequency periodically time varying controls and derive average solutions for system behavior we show how the pth order average formula can be used to construct open loop controls for point to point maneuvering of systems which require up to p 1 iterations of lie brackets to satisfy the lie algebra controllability rank condition in the cases p 2 3 we give algorithms for constructing these controls as a function of structure constants that define the control authority i e the actuator capability of the system the algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with o spl epsiv sup p accuracy in general exactly if the lie algebra is nilpotent the methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs

a fourth order adi method for semidiscrete parabolic equations abstract a fourth order fourstep adi method is described for solving the systems of ordinary differential equations which are obtained when a nonlinear parabolic initial boundary value problem in two dimensions is semi discretized the local time discretization error and the stability conditions are derived by numerical experiments it is demonstrated that the asymptotic fourth order behaviour does not degenerate if the time step increases to relatively large values also a comparison is made with the classical adi method of peaceman and rachford showing the superiority of the fourth order method in the higher accuracy region particularly in nonlinear problems