eliminating constraint drift in the numerical simulation of constrained dynamical systems by means of the udwadia kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift the basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion in order to avoid the drift typically experienced in constrained multibody simulations the embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense the formulation allows one to use a standard ode solver avoiding the need for iterative constraint stabilization the equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities the proposed method takes into account the finite word length of the computational environment and also accommodates possibly inconsistent initial conditions

exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

collocation software for boundary value differential algebraic equations we describe the methods and implementation of a general purpose code coldae this code can solve boundary value problems for nonlinear systems of semi explicit differential algebraic equations daes of index at most 2 fully implicit index 1 boundary value dae problems can be handled as well the code coldae is an extension of the package colnew colsys for solving boundary value odes the implemented method is piecewise polynomial collocation at gaussian points extended as needed by the projection method of ascher petzold for general semi explicit index 2 problems as well as for fully implicit index 1 problems we define a em selective projected collocation method and demonstrate its use the mesh selection procedure of colsys is modified for the case of index 2 constraints we also discuss shooting for initial guesses the power and generality of the code are demonstrated by examples

long time numerical solution of the navier stokes equations based on a sequential regularization formulation the sequential regularization method is a reformulation of the unsteady navier stokes equations from the viewpoint of constrained dynamical systems or the approximate helmholtz hodge projection in this paper we study the long time behavior of the sequential regularization formulation we give a uniform in time estimate between the solution of the reformulated system and that of the navier stokes equations we also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time a couple of long time flow examples are computed to demonstrate this method

regularization of higher index differential algebraic equations with rank deficient constraints in this paper we present several regularizations for higher index differential algebraic equations with rank deficient or singular constraints these types of problems arise for example in the solution of constrained mechanical systems when a mechanism s trajectory passes through or near a kinematic singularity we derive a class of regularizations for these problems which is based on minimization of the norm of the constraints the new regularizations are analogous to trust region methods of numerical optimization we give convergence results for the regularizations and present some numerical experiments which illustrate their effectiveness

multi multiplier ambient space formulations of constrained dynamical systems with an application to elastodynamics various formulations of the equations of motion for both finite and infinite dimensional constrained lagrangian dynamical systems are studied the different formulations correspond to different ways of enforcing constraints through multiplier fields all the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied the stability properties of the invariant set within its ambient space are shown to be different in each case we use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient space formulations we establish the well posedness of one formulation in the particular case of a homogeneous isotropic body subject to specified tractions on its boundary

symplectic integration of constrained hamiltonian systems by composition methods recent work reported in the literature suggests that for the long term integration of hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow in this paper the symplecticity of numerical integrators is investigated for constrained hamiltonian systems with holonomic constraints the following two results will be derived i it is shown that any first or second order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints based on this higher order methods can be derived by the same composition methods used for unconstrained problems ii leimkuhler and reich math comp 63 1994 pp 589 605 derived symplectic integrators based on dirac s reformulation of the constrained problem as an unconstrained hamiltonian system however although the unconstrained reformulation can be handled by direct application of any symplectic implicit runge kutta m

pseudospectral method for numerical solution of daes with an error estimation in e babolian m m hosseini reducing index and pseudospectral methods for differential algebraic equations appl math comput 140 2003 77 90 numerical solution of linear differential algebraic equations daes has been presented by pseudospectral method in this paper a new error estimation technique is proposed to pseudospectral method such that is well done for linear semi explicit daes when the daes has index 2 the obtained approximate solution can be improved by using the proposed error estimation technique furthermore with providing some examples the proposed error estimation method is dealt with numerically

stability criteria of random nonlinear systems and their applications stochastic differential equations sdes are widely adopted to describe systems with stochastic disturbances while they are not necessarily the best models in some specific situations this paper considers the nonlinear systems described by random differential equations rdes the notions and the corresponding criteria of noise to state stability asymptotic gain and asymptotic stability are proposed in the m th moment or in probability several estimation methods of stochastic processes are presented to explain the reasonability of the assumptions used in theorems as applications of stability criteria some examples about stabilization regulation and tracking are considered respectively a theoretical framework on stability of rdes is finally constructed which is distinguished from the existing framework of sdes

positive numerical integration methods for chemical kinetic systems abstract chemical kinetics conserves mass and renders nonnegative solutions a good numerical simulation would ideally produce mass balanced positive concentration vectors many time stepping methods are mass conservative however unconditional positivity restricts the order of a traditional method to one the projection method presented in this paper ensures mass conservation and positivity first a numerical approximation is computed with one step of a mass preserving traditional scheme if there are negative components the nearest vector in the reaction simplex is found by solving a quadratic optimization problem this vector is shown to better approximate the true solution a simpler version involves just one projection step and stabilizes the reaction simplex this technique works best when the underlying time stepping scheme favors positivity projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

on the computation of impasse points of quasi linear differential algebraic equations abstract we present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential algebraic equations our method combines a reduction step transforming the dae into a singular ode with an augmentation procedure inspired by numerical bifurcation theory singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory two numerical examples with physical relevance are given

noise to state stability for a class of random systems with state dependent switching this note is intended to investigate noise to state stability for random nonlinear systems with state dependent switching under some mild and easily verified conditions the existence of global solution to random switched systems can be proved based on a reasonable requirement for the random disturbance the criteria on noise to state stability of random switched systems are presented by the aid of single lyapunov function technique the reasonability of the obtained results is illustrated by using a mechanical model in random vibration environment

sequential regularization methods for nonlinear higher index daes sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints this paper extends an earlier work siam j numer anal 33 1996 pp 1921 1940 to nonlinear problems and to differential algebraic equations daes with an index higher than 2 rather than having one winning method this is a class of methods from which a number of variants are singled out as being particularly effective methods in certain circumstances we propose sequential regularization methods for index 2 and index 3 daes both with and without constraint singularities in the case of no constraint singularity we prove convergence results numerical experiments confirm our theoretical predictions and demonstrate the viability of the proposed methods the examples include constrained multibody systems

a fully explicit method for incompressible flow computation abstract a new formulation of the navier stokes equations is introduced to solve incompressible flow problems it keeps the benefits of the penalty method that is velocity and pressure can be obtained separately and no pressure poisson equation is involved unlike the penalty method the formulation is more stable or less stiff and then explicit time integration can be applied for easy implementation no linear or nonlinear system need be solved in the method in the case that a large number of time steps are needed a parallelization based on domain decomposition is applied to reduce the computational time with the explicit time integration the parallel implementation and its message passing are very simple as well

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

reducing index method for differential algebraic equations with constraint singularities this paper extends an earlier work appl math comput 140 2003 77 to differential algebraic equations with constraint singularities numerical solution of these problems is considered by pseudospectral method with domain decomposition and by giving a condition under which the general linear form problem can easily be transformed to the index reduced form by a simple formulation furthermore with providing some examples the aforementioned cases are dealt with numerically

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part i low order methods for two model problems and nonlinear elastodynamics abstract we present in this paper the development of a class of time stepping algorithms for nonlinear elastodynamics that exhibits the controllable numerical dissipation in the high frequency range required for the robust solution of the resulting numerically stiff systems to motivate and illustrate better the developments in this general case we present first the formulation and analysis of these methods for two simple model problems namely we consider a nonlinear elastic spring mass system and a simplified model of thin elastic beams as it is discussed in detail in this paper the conservation by the numerical algorithm of the momenta and corresponding relative equilibria of these characteristic hamiltonian systems with symmetry is of the main importance these conservation properties lead for a fixed and finite time step to a correct qualitative picture of the phase space where the discrete dynamics takes place even in the presence of the desired and controlled numerical dissipation of the energy this situation is contrasted with traditional dissipative numerical schemes which are shown through rigorous analyses to not only lose their dissipative character in the general nonlinear range but also the aforementioned conservation properties thus leading to a qualitatively distorted approximation of the phase dynamics the key for a successful algorithm in this context is the incorporation of the numerical dissipation in the internal modes of the motion while not affecting the group motions of the systems the algorithms presented in this work accomplish these goals the focus in this first part is given to first order methods representative numerical simulations ranging from applications in nonlinear structural dynamics to nonlinear continuum three dimensional elastodynamics are presented in the context of the finite element method to illustrate these ideas and results

constraint preserving integrators for general nonlinear higher index daes in the last few years there has been considerable research on numerical methods for differential algebraic equations daes f x x t 0 where f x is identically singular the index provides one measure of the singularity of a dae most of the numerical analysis literature on daes to date has dealt with daes with indices no larger than three even in this case the systems were often assumed to have a special structure recently a numerical method was proposed that could in principle be used to integrate general unstructured higher index solvable daes however that method did not preserve constraints this paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index daes

global validity of the boltzmann equation for two and three dimensional rare gas in vacuum erratum and improved result we point out an error in our earlier papers 1 and 2 and present a more direct and natural proof which although based on the same physical ideas of the previous ones saves and actually improves the validity results for the boltzmann equation given in 1 and 2

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

adomian decomposition method for solution of differential algebraic equations solutions of differential algebraic equations is considered by adomian decomposition method in e babolian m m hosseini reducing index and spectral methods for differential algebraic equations j appl math comput 140 2003 77 and m m hosseini an index reduction method for linear hessenberg systems j appl math comput in press an efficient technique to reduce index of semi explicit differential algebraic equations has been presented in this paper adomian decomposition method is applied to reduced index problems the scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

stabilization of daes and invariant manifolds many methods have been proposed for the stabilization of higher index differential algebraic equations daes such methods often involve constraint differentiation and problem stabilization thus obtaining a stabilized index reduction a popular method is baumgarte stabilization but the choice of parameters to make it robust is unclear in practice here we explain why the baumgarte method may run into trouble we then show how to improve it we further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature our approach is to i consider stabilization of odes with invariants ii discretize the stabilizing term in a simple way generally different from the ode discretization and iii use orthogonal projections whenever possible the best methods thus obtained are related to methods of coordinate projection we discuss them and make concrete algorithmic suggestions

analysis of a sequential regularization method for the unsteady navier stokes equations the incompressibility constraint makes navier stokes equations difficult a reformulation to a better posed problem is needed before solving it numerically the sequential regularization method srm is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods in the paper we study the existence and uniqueness for the solution of the srm and provide a simple proof of the convergence of the solution of the srm to the solution of the navier stokes equations we also give error estimates for the time discretized srm formulation

corrections to noise to state stability for a class of random systems with state dependent switching there are two errors in the proof of 1 th 1 for which a correction is provided and correspondingly a redundant condition of theorems 1 and 2 is removed

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

global validity of the boltzmann equation for a three dimensional rare gas in vacuum we consider a system ofn hard disks in 2 in the boltzmann grad limit i e n d 0 n d 1 0 whered is the diameter of the disks if is sufficiently small and if the joint distribution densities factorize at time zero we prove that the time evolved one particle distribution converges for all times to the solution of the boltzmann equation with the same initial datum

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

regularization of index 1 differential algebraic equations with rank deficient constraints abstract in this paper we present a regularization for semiexplicit index 1 differential algebraic equations with rank deficient or singular constraints we consider those problems for which the solution is well defined through the singularity we give convergence results for the regularization applied to linear daes and present some numerical experiments which illustrate its effectiveness

kinetic limits for a class of interacting particle systems we study one dimensional particle systems in which particles travel as independent random walks and collide stochastically the collision rates are chosen so that each particle experiences finitely many collisions per unit time we establish the kinetic limit and derive the discrete boltzmann equation for the macroscopic particle density

a predicted sequential regularization method for index 2 hessenberg daes the sequential regularization method srm is a dynamic iterative method for the numerical solution of higher index differential algebraic equations daes the srm has the advantage of being based on a regularized problem that is less stiff than those produced by standard regularization methods consequently nonstiff integrators may be used making the srm a competitive alternative to popular integrators in past work the number of srm iterations was taken to be roughly equal to the order of the numerical method used in each dynamic iteration in this paper we propose a predicted srm psrm that reduces the number of iterations in each dynamic iteration to one we give a new error analysis for explicit runge kutta methods applied to linear index 2 hessenberg daes with or without singularities we also give numerical examples to confirm the predicted convergence rates for the psrm extrapolation formulas and methods based on the differential part of the daes serve as a predictor and the srm iteration serves as a corrector implementation of higher order schemes for the psrm makes use of continuous extensions of runge kutta methods in particular we give a prediction scheme for the algebraic variable at intermediate stage points that suppresses order reduction in the differential variable near a singularity moreover the srm psrm provides new insight into operator splitting and fast convergence rates for waveform relaxation

the alpha method direct transcription in path constrained dynamic optimization numerically discretized dynamic optimization problems having active inequality and equality path constraints that along with the dynamics induce locally high index differential algebraic equations often cause the optimizer to fail in convergence or to produce degraded control solutions in many applications regularization of the numerically discretized problem in direct transcription schemes by perturbing the high index path constraints helps the optimizer to converge to useful control solutions for complex engineering problems with many constraints it is often difficult to find effective nondegenerate perturbations that produce useful solutions in some neighborhood of the correct solution in this paper we describe a numerical discretization that regularizes the numerically consistent discretized dynamics and does not perturb the path constraints for all values of the regularization parameter the discretization remains numerically consistent with the dynamics and the path constraints specified in the original problem the regularization is quantifiable in terms of time step size in the mesh and the regularization parameter for fully regularized systems the scheme converges linearly in time step size the method is illustrated with examples

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

sequential regularization methods for simulating mechanical systems with many closed loops the numerical simulation problem of large multibody systems has often been treated in two separate stages i the forward dynamics problem for computing system accelerations from given force functions and constraints and ii the numerical integration problem for advancing the state in time for the forward dynamics problem algorithms have been given with optimal linear complexity in the number of bodies in case the system topology does not contain many closed loops but the interaction between these two stages can be important using explicit time integration schemes we propose a sequential regularization method srm that has a linear complexity in the number of bodies per time step even in the presence of many closed loops the method also handles certain types of constraint singularity

reducing index and pseudospectral methods for differential algebraic equations in this paper numerical solution of linear differential algebraic equations is considered by pseudospectral method also an index reduction technique for them is suggested for hessenberg index 2 system the index reduction technique is defined and its use is demonstrated we give a condition under which the general linear form problem can be easily transformed to the index reduced form by a simple formulation furthermore with providing some examples the aforementioned cases are dealt with numerically

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

differential algebraic equations as stiff ordinary differential equations this paper deals with the relation between differential algebraic equations daes and certain stiff odes and their respective discretizations by implicit runge kutta methods for that purpose for any dae a singular perturbed ode is constructed such that the dae is its reduced problem and the solution of the ode converges in some sense to that of the dae thus the dae can be interpreted as an infinitely stiff ode an analysis of the discretization error of this singular perturbed system gives insight into the relationship of order reduction phenomena observed for stiff odes to that for daes analysis of a general class of singularly perturbed problems and their discretizations is not attempted however the technique of treating singularly perturbed problems and daes in a unified way is new and can possibly be applied to other systems and their discretizations as well since asymptotic expansions are not used but an approach similar to the ones used in b convergence theory is applied one can derive erro

a time integration algorithm for flexible mechanism dynamics the dae α method this paper introduces a new family of second order methods for solving the index 2 differential algebraic equations daes of motion for flexible mechanism dynamics these methods which extend the methods for odes of structural dynamics to daes possess numerical dissipation that can be controlled by the user convergence and stability analysis is given and verifies that the dae methods introduce no additional oscillations and preserve the stability of the underlying ode system convergence of the newton iteration which can be a source of difficulty in solving nonlinear oscillatory systems with large stepsizes is achieved via a coordinate split modification to the newton iteration numerical results illustrate the effectiveness of the new methods for simulation of flexible mechanisms

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

conserving first integrals under discretization with variable step size integration procedures abstract it is well known that the application of one step or linear multistep methods to an ordinary differential equation with first integrals will destroy the conserving quantities with the use of stabilization techniques similar to ascher chin reich numer math 67 1997 131 149 we derive stabilized variants of our original problem we show that variable step size one step and linear multistep methods applied to the stabilized equation will reproduce that phase portrait correctly in particular this technique will conserve first integrals over an infinite time interval within the local error of the used method

a stochastic model associated with enskog equation and its kinetic limit we examine a system of particles in which the particles travel deterministically in between stochastic collisions the collisions are elastic and occur with probability d when two particles are at a distance when the number of particles n goes to infinity and n d goes to a nonzero constant we show that the particle density converges to a solution of the enskog equation

time step size independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations we propose a simple preconditioning for the equations of motion of constrained mechanical systems in index three form the scaling transformation is applied to the displacement velocity multiplier and to the reduced displacement multiplier forms the analysis of the transformed system shows that conditioning and sensitivity to perturbations become independent of the time step size as in the case of well behaved ordinary differential equations the new scaling transformation is simple to implement and does not require the rewriting of the system equations as other approaches do the theoretical analysis is confirmed by numerical examples

stability of computational methods for constrained dynamics systems many methods have been proposed for numerically integrating the differential algebraic systems arising from the euler lagrange equations for constrained motion these are based on various problem formulations and discretizations we offer a critical evaluation of these methods from the standpoint of stability nconsidering a linear model we first give conditions under which the differential algebraic problem is well conditioned this involves the concept of an essential underlying ode we review a variety of reformulations which have been proposed in the literature and show that most of them preserve the well conditioning of the original problem then we consider stiff and nonstiff discretizations of such reformulated models in some cases the same implicit discretization may behave in a very different way when applied to different problem formulations acting as a stiff integrator on some formulations and as a nonstiff integrator on others we present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense

numerical integration of constrained hamiltonian systems using dirac brackets we study the numerical properties of the equations of motion of constrained systems derived with dirac brackets this formulation is compared with one based on the extended hamiltonian as concrete examples a pendulum in cartesian coordinates and a chain molecule are treated

methods for constraint violation suppression in the numerical simulation of constrained multibody systems a comparative study abstract multibody systems are frequently modeled as constrained systems and the arising governing equations incorporate the closing constraint equations at the acceleration level one consequence of accumulation of integration truncation errors is the phenomenon of violation of the lower order constraint equations by the numerical solutions to the governing equations the constraint drift usually tends to increase in time and may spoil reliability of the simulation results in this paper a comparative study of three methods for constraint violation suppression is presented the popular baumgarte s constraint violation stabilization method a projective scheme for constraint violation elimination and a novel scheme patterned after that proposed recently by braun and goldfarb d j braun m goldfarb eliminating constraint drift in the numerical simulation of constrained dynamical systems comput meth appl mech engrg 198 2009 3151 3160 the methods are confronted with respect to simplicity in applications numerical effectiveness and influence on accuracy of the constraint consistent motion

singularity crossing phenomena in daes a two phase fluid flow application case study this paper analyzes the existence of smooth trajectories through singular points of differential algebraic equations or daes arising from traveling wave solutions of a degenerate convection diffusion model the dae system can be written in the quasilinear form a x x b x in this setting singularities are displayed when the matrix a x undergoes a rank change the singular hypersurface may be smoothly crossed by trajectories in a finite time if x is a geometric singularity satisfying certain directional conditions the basis of our analysis is a two phase fluid flow model in one spatial dimension with dissipative mechanism involved

a cluster expansion approach to a one dimensional boltzmann equation a validity result we consider a stochastic particle system on the line and prove that when the number of particles diverges and the probability of a collision is suitably rescaled the system is well described by a one dimensional boltzmann equation the result holds globally in time without any smallness assumption

boltzmann grad limits for stochastic hard sphere models we study a d dimensional stochastic particle system in which the particles travel deterministically in between stochastic collisions the collisions are elastic and occur with a probability of order when two particles are at a distance less than when the number of particles n goes to infinity and n d 1 goes to a nonzero constant we show that the particle density converges to a solution of the boltzmann equation provided that d 1

computing lyapunov exponents on a stiefel manifold the problem of numerical computation of a few lyapunov exponents les of finite dimensional dynamical systems is considered from the viewpoint of the differential geometry of stiefel manifolds whether one computes one many or all les of a continuous dynamical system by time integration discrete or continuous orthonormalization is essential for stable numerical integration a differential geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a stiefel manifold however the stiefel manifold is not in general an attracting submanifold of the ambient euclidean space it is a constraint manifold with a weak numerical invariant new numerical algorithms for this problem are then designed which use the fiber bundle characterization of these manifolds this framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew symmetry property numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few les which preserve orthonormality to machine accuracy this idea is also taken a step further by making the stiefel manifold an attracting invariant manifold in which case standard explicit runge kutta algorithms can be used this leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization these class of methods should be particularly effective for computing a few les for large dimension dynamical systems the new schemes are straightforward to implement a test case is presented for illustration and an example from dynamical systems is presented where a few les are computed for an array of coupled oscillators

convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints the equations of motion of mechanical multibody systems with algebraic constraints are of index 3 and therefore not directly solvable by standard ode or dae methods reducing the index by different

sequential regularization methods for higher index daes with constraint singularities the linear index 2 case standard stabilization techniques for higher index differential algebraic equations daes often involve elimination of the algebraic solution components this may not work well if there are singularity points where the constraint jacobian matrix becomes rank deficient this paper proposes instead a sequential regularization method srm a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components for linear index 2 daes we consider both initial value problems ivps and boundary value problems bvps the convergence of the srm is described and proved in detail we believe that this is the first convergence proof for any method for daes with this type of constraint singularities moreover the regularization parameter in our method is not necessarily very small so the srm is an important improvement over usual regularization methods various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out

completions of nonlinear dae flows based on index reduction techniques and their stabilization differential algebraic equations daes define a differential equation on a manifold a number of ways have been developed to numerically solve some classes of daes motivated by problems in control theory numerical simulation and the use of general purpose modeling environments recent research has considered the embedding of the dae solutions of a general dae into the solutions of an ode where the added dynamics have special properties this paper both provides new results on the linear time varying case and considers the important nonlinear case

a derivation of the broadwell equation we consider a stochastic system of particles in a two dimensional lattice and prove that under a suitable limit i e n e 0 ne2 const wheren is the number of particles and e is the mesh of the lattice the one particle distribution function converges to a solution of the two dimensional broadwell equation for all times for which the solution of this equation exists propagation of chaos is also proven

a sequential regularization method for time dependent incompressible navier stokes equations the objective of the paper is to present a method called the sequential regularization method srm for the nonstationary incompressible navier stokes equations from the viewpoint of regularization of differential algebraic equations daes and to provide a way to apply a dae method to partial differential algebraic equations pdaes the srm is a functional iterative procedure it is proved that its convergence rate is o epsilon m where m is the number of the srm iterations and epsilon is the regularization parameter the discretization and implementation issues of the method are considered in particular a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes it appears that the srm formulation is new in the navier stokes context unlike other regularizations or pseudocompressibility methods in the navier stokes context the regularization parameter epsilon in the srm need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any epsilon hence the stability condition is independent of epsilon even for explicit time discretization numerical experiments are given to verify our theoretical results

adomian decomposition method for solution of nonlinear differential algebraic equations in m m hosseini adomian decomposition method with chebyshev polynomials appl math comput in press an efficient modification of the adomian decomposition method was presented by using chebyshev polynomials also in m m hosseini adomian decomposition method for solution of differential algebraic equations j comput appl math in press solution of linear differential algebraic equations was considered by adomian decomposition method in this paper standard and modified adomian decomposition methods are applied to non linear differential algebraic equations the schemes are tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

large deviations from a kinetic limit we study a one dimensional particle system in which particles travel deterministically in between stochastic collisions as the total number of particles tends to infinity the empirical density converges to a solution of a discrete boltzmann equation we establish the large deviation principle for the convergence with a rate function that is given by a variational formula some of the properties of the rate function are discussed and a nonvariational expression for the rate function is given

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

system of linear ordinary differential and differential algebraic equations and pseudo spectral method in this paper first we introduce briefly pseudo spectral method to solve linear ordinary differential equations odes and then extend it to solve a system of linear odes and then differential algebraic equations daes furthermore because of appropriate choice of chebyshev gauss raudo points we will show that this method can be used to solve a dae whenever some coefficient functions in constraint are not analytic by providing some numerical examples

an index reduction method for linear hessenberg systems in e babolian m m hosseini reducing index and pseudospectral methods for differential algebraic equations appl math comput 140 2003 77 90 a reducing index method has been proposed for some cases of semi explicit daes differential algebraic equations in this paper this method is generalized to more cases also it is focused on hessenberg index 2 systems and proposed reduction index method will be illustrated for this problem the hessenberg system and its obtained reduced index system are numerically solved through pseudospectral method in addition aforementioned methods will be considered by one example

convergence of nonconvergent irk discretizations of optimal control problems with state inequality constraints it has been observed that optimization codes are sometimes able to solve inequality state constrained optimal control problems with discretizations which do not converge when used as integrators on the constrained dynamics understanding this phenomenon could lead to a more robust design for direct transcription codes as well as better test problems this paper examines how this phenomenon can occur the difference between solving index 3 differential algebraic equations daes using the trapezoid method in the context of direct transcription for optimal control problems and a straightforward implicit runge kutta irk formulation of the same trapezoidal discretization is analyzed it is shown through numerical experience and theory that the two can differ as much as o 1 h3 in the control the optimization can use a small sacrifice in the accuracy of the states in the early stages of the trapezoidal method to produce better accuracy in the control whereas more precise solutions converge to an incorrect solution convergence independent of the index of the constraints is then proven for one class of problems the theoretical results are used to explain computational observations

on some difference schemes for singular singularly perturbed boundary value problems singularly perturbed boundary value ordinary differential problems are considered where the problem defining the reduced solution is singular for numerical approximation families of symmetric difference schemes which are equivalent to certain collocation schemes based on gauss and lobatto points are used convergence results previously obtained for the regular singularly perturbed case are extended while gauss schemes are extended with no change lobatto schemes require a small modification in the mesh selection procedure with meshes as prescribed in the text highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost this is demonstrated by examples

completions of implicitly defined linear time varying vector fields the stabilization of constraints through such techniques as baumgarte stabilization has been used in the simulation community for some time this and a number of control problems can be viewed as either extending or modifying a vector field off of some manifold generally these approaches required the equations to have a special structure motivated by numerical simulation there has recently been new progress on doing this stabilization in numerically robust ways for larger classes of systems in this paper we point out how earlier linear time invariant results do not immediately apply to the linear time varying case and then analyze the linear time varying case

numerical solution of linear differential algebraic equations in appl math comput 140 2003 77 and appl math comput in press a reducing and modified reducing index methods have been proposed for some cases of semi explicit daes differential algebraic equations in this paper these methods are generalized to more cases also numerical implementation of generalized methods will be presented through pseudospectral method with domain decomposition in addition aforementioned methods will be considered by some examples

regularized numerical integration of multibody dynamics with the generalized α method this paper discusses the consistent regularization property of the generalized a method when applied as an integrator to an initial value high index and singular differential algebraic equation model of a multibody system the regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take the regularization involves increase of the smallest singular values of the ill conditioned jacobian of the discretization and is different from baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter this regularization also helps where pre conditioning the jacobian by scaling is of limited effect for example when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present the feed forward control in kane s equation models is additionally considered in the numerical examples to illustrate the effect of regularization the discretization presented in this work is adopted to the first order dae system unlike the original method which is intended for second order systems for its a stability and same order of accuracy for positions and velocities

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability