specialized runge kutta methods for index 2 differential algebraic equations we consider the numerical solution of systems of semi explicit index 2 differential algebraic equations daes by methods based on runge kutta rk coefficients for nonstiffly accurate rk coefficients such as gauss and radau ia coefficients the standard application of implicit rk methods is generally not superconvergent to reestablish superconvergence projected rk methods and partitioned rk methods have been proposed in this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage moreover symmetry of gauss methods is preserved the main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish we call these methods specialized runge kutta methods for index 2 daes srk dae2

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

discrete time port hamiltonian systems a definition based on symplectic integration abstract we introduce a new definition of discrete time port hamiltonian ph systems which results from structure preserving discretization of explicit ph systems in time we discretize the underlying continuous time dirac structure with the collocation method and add discrete time dynamics by the use of symplectic numerical integration schemes the conservation of a structural discrete time energy balance expressed in terms of the discrete time dirac structure extends the notion of symplecticity of geometric integration schemes to open systems we discuss the energy approximation errors in the context of the presented definition and show that their order for linear ph systems is consistent with the order of the numerical integration scheme implicit gauss legendre methods and lobatto iiia iiib pairs for partitioned systems are examples for integration schemes that are covered by our definition the statements on the numerical energy errors are illustrated by elementary numerical experiments

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

partitioned runge kutta methods for separable hamiltonian problems separable hamiltonian systems of differential equations have the form dp dt dh dq dq dt dh dp with a hamiltonian function h that satisfies h t p k q t and v are respectively the kinetic and potential energies we study the integration of these systems by means of partitioned runge kutta methods i e by means of methods where different runge kutta tableaux are used for the p and q equations we derive a suffi cient and almost necessary condition for a partitioned runge kutta method to be canonical i e to conserve the symplectic structure of phase space thereby reproducing the qualitative properties of the hamiltonian dynamics we show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

variational collision integrator for polymer chains the numerical simulation of many particle systems e g in molecular dynamics often involves constraints of various forms we present a symplectic integrator for mechanical systems with holonomic bilateral and unilateral contact constraints the latter being in the form of a non penetration condition the scheme is based on a discrete variant of hamilton s principle in which both the discrete trajectory and the unknown collision time are varied cf r fetecau j marsden m ortiz m west nonsmooth lagrangian mechanics and variational collision integrators siam j appl dyn syst 2 2003 381 416 as a consequence the collision event enters the discrete equations of motion as an unknown that has to be computed on the fly whenever a collision is imminent the additional bilateral constraints are efficiently dealt with employing a discrete null space reduction including a projection and a local reparametrisation step which considerably reduces the number of unknowns and improves the condition number during each time step as compared to a standard treatment with lagrange multipliers we illustrate the numerical scheme with a simple example from polymer dynamics a linear chain of beads and test it against other standard numerical schemes for collision problems

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

on rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction this paper deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements we analyze the structure of the equations of motion and show that a differential algebraic system of index 1 with a singular perturbation term needs to be solved in the limit case the index may jump to index 3 and thus renders the time integration into a difficult problem for the time integration we apply rosenbrock methods and study their convergence behavior for a test problem which highlights the importance of the well known scholz conditions for this problem class numerical tests demonstrate that such linear implicit methods are an attractive alternative to established time integration methods in structural dynamics we also combine the simulation of nonlinear materials with a model reduction step in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications

preserving poisson structure and orthogonality in numerical integration of differential equations we consider the numerical integration of two types of systems of differential equations we first consider hamiltonian systems of differential equations with a poisson structure we show that symplectic runge kutta methods preserve this structure when the poisson tensor is constant using nonlinear changes of coordinates this structure can also be preserved for nonconstant poisson tensors as exemplified on the euler equations for the free rigid body we also consider orthogonal flows and the closely related class of isospectral flows to numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential algebraic equations daes and of integrating the system with a special combination of a particular class of runge kutta methods this approach requires only matrix matrix products and can preserve geometric properties of the flow such as reversibility

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

the nose poincare method for constant temperature molecular dynamics we present a new extended phase space method for constant temperature canonical ensemble molecular dynamics our starting point is the hamiltonian introduced by nose to generate trajectories corresponding to configurations in the canonical ensemble using a poincare time transformation we construct a hamiltonian system with the correct intrinsic timescale and show that it generates trajectories in the canonical ensemble our approach corrects a serious deficiency of the standard change of variables nose hoover dynamics which yields a time reversible system but simultaneously destroys the hamiltonian structure a symplectic discretization method is presented for solving the nose poincare equations the method is explicit and preserves the time reversal symmetry in numerical experiments it is shown that the new method exhibits enhanced stability when the temperature fluctuation is large extensions are presented for nose chains holonomic constraints and rigid bodies

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

symplectic partitioned runge kutta methods for constrained hamiltonian systems this article deals with the numerical treatment of hamiltonian systems with holonomic constraints a class of partitioned runge kutta methods consisting of the couples of s stage lobatto iiia and lobatto iiib methods has been discovered to solve these problems efficiently these methods are symplectic preserve all un derlying constraints and are superconvergent with order 2s 2 for separable hamiltonians of the form h q p frac 1 2 p t m 1 p u q the rattle algorithm based on the verlet method was up to now the only known symplectic method preserving the constraints in fact this method turns out to be equivalent to the 2 stage lobatto iiia iiib method of order 2 numerical examples have been performed which illustrate the theoretical results

some new integrable nonlinear evolution equations and darboux transformation two classes of new nonlinear evolution equations are derived with the aid of a 4 4 matrix spectral problem with three potentials a darboux transformation for the first nontrivial nonlinear evolution equation is given based on the gauge transformation between the corresponding 4 4 matrix spectral problems as an application of the darboux transformation some explicit solutions of the nontrivially nonlinear evolution equation are obtained

the discrete null space method for the energy consistent integration of constrained mechanical systems part i holonomic constraints the present work deals with energy consistent time stepping schemes for finite dimensional mechanical systems with holonomic constraints the proposed procedure is essentially based upon the following steps firstly the index three differential algebraic equations corresponding to the constrained mechanical system are directly discretized secondly the discrete lagrange multipliers are eliminated by using a discrete null space matrix in many cases it is feasible to further reduce the number of unknowns by employing specific reparametrizations the proposed method entails a number of advantageous features such as size reduction and improved conditioning of the resulting system of algebraic equations it is shown that the newly developed method is well suited for both open loop and closed loop multibody systems

generalized boltzmann equation for lattice gas automata in this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi detailed balance starting from a coupled bbgky hierarchy for then particle distribution functions cluster expansion techniques are used to derive approximate kinetic equations in zeroth approximation the standard nonlnear boltzmann equation is obtained the next approximation yields the ring kinetic equation similar to that for hard sphere systems describing the time evolution of pair correlations the ring equation is solved to determine the nonvanishing pair correlation functions in equilibrium for two models that violate semidetailed balance one is a model of interacting random walkers on a line the other one is a two dimensional fluid type model on a triangular lattice the numerical predictions agree very well with computer simulations

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

a time reversible variable stepsize integrator for constrained dynamics this article considers the design and implementation of variable timestep methods for simulating holonomically constrained mechanical systems symplectic variable stepsizes are briefly discussed and we consider time reparameterization techniques employing a time reversible symmetric integration method to solve the equations of motion we give several numerical examples including a simulation of an elastic inextensible unshearable rod undergoing large deformations and collisions with the sides of a bounding box numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation

on higher order semi explicit symplectic partitioned runge kutta methods for constrained hamiltonian systems in this paper we generalize the class of explicit partitioned runge kutta prk methods for separable hamiltonian systems to systems with holonomic constraints for a convenient analysis of such schemes we first generalize the backward error analysis for systems in bbb r m to systems on manifolds embedded in bbb r m by applying this analysis to constrained prk methods we prove that such methods will in general suffer from order reduction as well known for higher index differential algebraic equations however this order reduction can be avoided by a proper modification of the standard prk methods this modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant we also give a numerical comparison of several second fourth and sixth order methods

on the application of a generalized dressing method to the integration of variable coefficient coupled hirota equations based on the generalized dressing method we propose two variable coefficient coupled hirota equations and derive their lax pairs the generalized dressing method is applied to study these variable coefficient coupled hirota equations from which explicit solutions of the two equations and their reductions are constructed

alternative integration methods for problems in structural dynamics abstract runge kutta methods for the time integration of the equations of motion in structural dynamics are presented the methods belong to a class called sdirk singly diagonally implicit runge kutta methods the computational cost needed per step is comparable to newmark methods with damping the methods proposed are l stable which means that they instantly damp out the higher modes in the solution the numerical dissipation can be controlled by a parameter both second and third order methods are presented some characteristics of the methods are compared with the newmark method with damping

linear stability of partitioned runge kutta methods we study the linear stability of partitioned runge kutta prk methods applied to linear separable hamiltonian odes and to the semidiscretization of certain hamiltonian pdes we extend the work of jay and petzold highly oscillatory systems and periodic stability preprint 95 015 army high performance computing research center stanford ca 1995 by presenting simplified expressions of the trace of the stability matrix tr m s for the lobatto iiia iiib family of symplectic prk methods by making the connection to pade approximants and continued fractions we study the asymptotic behavior of tr m s omega as a function of the frequency omega and stage number s

the numerical solution of higher index differential algebraic equations by implicit methods this paper studies the order stability and convergence properties of implicit runge kutta irk methods applied to differential algebraic systems with index greater than one these methods do not in general attain the same order of accuracy for higher index differential algebraic systems as they do for index 1 systems or for purely differential systems necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems irk methods applied to nonlinear semi explicit index 2 systems are studied and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems finally some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff errors on the solution

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

on multisymplecticity of partitioned runge kutta methods previously it has been shown that discretizing a multi hamiltonian pde in space and time with partitioned runge kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law however these previous studies use the same partitioning of the variables into two parts in both space and time this gives rise to a large number of cases to be considered each with its own set of conditions to be satisfied we present here a much simpler set of conditions covering all of these cases where the variables are partitioned independently in space and time into an arbitrary number of parts in general it is not known when such a discretization of a multi hamiltonian pde will give rise to a well defined numerical integrator however a numerical integrator that is explicit will typically be well defined in this paper we give sufficient conditions on a multi hamiltonian pde for a lobatto iiia iiib discretization in space to give rise to explicit odes and an algorithm for constructing these odes

some comments on dae theory for irk methods and trajectory optimization it has been observed elsewhere in the literature that the activation of constraints in a trajectory optimization problem can lead to higher index daes several existing codes can handle a number of these constrained problems in this note we will discuss why the situation is more complex than just saying a higher index dae occurs the discussion is in the context of a specific industrial code socs but the observations made here have relevance for a number of methods and have implications for what types of test problems a code should be tested on

collocation methods for differential algebraic equations of index 3 this article give sharp convergence results for stiffly accurate collocation methods as applied to differential algebraic equations dae s of index 3 in hessenberg form proving partially a conjecture of hairer lubich and roche

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

semi implicit reversible algorithms for rigid body rotational dynamics this paper presents two semi implicit algorithms based on splitting methodology for rigid body rotational dynamics the first algorithm is a variation of partitioned runge kutta prk methodology that can be formulated as a splitting method the second algorithm is akin to a multiple time stepping scheme and is based on modified crouch grossman mcg methodology which can also be expressed as a splitting algorithm these algorithms are second order accurate and time reversible however they are not poisson integrators i e non symplectic these algorithms conserve some of the first integrals of motion but some others are not conserved however the fluctuations in these invariants are bounded over exponentially long time intervals these algorithms exhibit excellent long term behavior because of their reversibility property and their approximate poisson structure preserving property the numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the simo wong algorithm newmark algorithm discrete moser veselov algorithm lewis simo algorithm and the liemid ea algorithm

energy conservation and symplectic properties of continuous finite element methods for hamiltonian systems using continuous finite element methods of ordinary differential equation we can get the first second third order continuous finite element methods for linear hamiltonian systems which are symplectic and conserve energy in addition the second order continuous finite element methods for nonlinear hamiltonian systems are approximately symplectic on third order accuracy as well as they conserve energy the numerical results are in agreement with theory

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

global modified hamiltonian for constrained symplectic integrators we prove that the numerical solution of partitioned runge kutta methods applied to constrained hamiltonian systems e g the rattle algorithm or the lobatto iiia iiib pair is formally equal to the exact solution of a constrained hamiltonian system with a globally defined modified hamiltonian this property is essential for a better understanding of their longtime behaviour as an illustration the equations of motion of an unsymmetric top are solved using a parameterization with euler parameters

an inverse scattering scheme for the regularized long wave equation the regularized long wave rlw equation was proposed as an alternative model to the korteweg de vries kdv equation to describe small amplitude long waves in shallow water however unlike the kdv equation which is exactly solvable by the inverse scattering method the rlw equation is deemed not to be completely integrable in this paper an inverse scattering scheme is described for solving the rlw equation which is based on the dressing method of zakharov and shabat significantly the compatibility of the dressed operators leads to two governing equations which taken together reduce to the rlw equation one of these equations links the rlw equation to the completely integrable shallow water wave asww equation due to ablowitz et al the second is a linear wave constraint that signals the nonintegrability of the rlw equation and prohibits the existence of multisoliton solutions contrary to results already reported in the literature the proposed scheme associates the rlw equation with a third order scattering problem moreover as a special case of the rlw scheme we obtain an inverse scattering procedure for the integrable asww equation its formulation via the dressing method would appear to be given here for the first time explicit solutions are constructed for both equations using the respective inverse scattering schemes

symmetric multistep methods for constrained hamiltonian systems a method of choice for the long time integration of constrained hamiltonian systems is the rattle algorithm it is symmetric symplectic and nearly preserves the hamiltonian but it is only of order two and thus not efficient for high accuracy requirements in this article we prove that certain symmetric linear multistep methods have the same qualitative behavior and can achieve an arbitrarily high order with a computational cost comparable to that of the rattle algorithm

conservation properties of a time fe method part iii mechanical systems with holonomic constraints a galerkin based discretization method for index 3 differential algebraic equations pertaining to finite dimensional mechanical systems with holonomic constraints is proposed in particular the mixed galerkin mg methodis introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained hamiltonian system namely total energy and angular momentum in addition to that the constraints on the configuration level and on the velocity momentum level are fulfilled exactly the application of the mg method to specific mechanical systems such as the pendulum rigid body dynamics and the coupled motion of rigid and flexible bodies is presented related numerical examples are investigated to evaluate the numerical performance of the mg 1 and mg 2 method

smoothed dynamics of highly oscillatory hamiltonian systems abstract we consider the numerical treatment of hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential such systems arise e g in molecular dynamics simulations and the spatial discretization of hamiltonian partial differential equations since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size the numerical integration of such systems provides a challenging task it has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential this approach has e g been successfully applied to the bond stretching in molecular dynamics simulations in other cases such as the bond angle bending this methods fails due to the introduced rigidity here we give a careful analysis of the analytical problem by means of a smoothing operator this will lead us to the notion of the smoothed dynamics of a highly oscillatory hamiltonian system based on our analysis we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high frequency components in the solutions and thus allowing for larger time steps the new constrained formulation is hamiltonian and can be discretized by the well known shake method

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

symplectic integration of constrained hamiltonian systems a hamiltonian system in potential form h q p ptmp 2 f q subject to smooth constraints on q can be viewed as a hamiltonian system on a manifold but numerical computations must be performed in rn in this paper methods which reduce hamiltonian differential algebraic equations to odes in euclidean space are examined the authors study the construction of canonical parametrizations or local charts as well as methods based on the construction of ode systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold in each case a hamiltonian system of ordinary differential equations is produced the stability of the constraint invariants and the behavior of the original hamiltonian along solutions are investigated both numerically and analytically

beyond conventional runge kutta methods in numerical integration of odes and daes by use of structures and local models there are two parts in this paper in the first part we consider an overdetermined system of differential algebraic equations daes we are particularly concerned with hamiltonian and lagrangian systems with holonomic constraints the main motivation is in finding methods based on gauss coefficients preserving not only the constraints symmetry symplecticness and variational nature of trajectories of holonomically constrained hamiltonian and lagrangian systems but also having optimal order of convergence the new class of s s gauss lobatto specialized partitioned additive runge kutta spark methods uses greatly the structure of the daes and possesses all desired properties in the second part we propose a unified approach for the solution of ordinary differential equations odes mixing analytical solutions and numerical approximations the basic idea is to consider local models which can be solved efficiently for example analytically and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction in order to preserve also symmetry we define the new class of symmetrized runge kutta methods with local model srklm

a numerical test of long time stability for rigid body integrators summary in the context of hamiltonian odes a necessary condition for an integrator to be symplectic or conjugate symplectic is that it nearly preserves the exact hamiltonian this paper introduces a numerical test of this necessity for rigid body methods it turns out that several rigid body integrators proposed in literature fail this test hence these integrators should be used with caution for long time simulation copyright 2011 john wiley sons ltd

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

the applicability of constrained symplectic integrators in general relativity the purpose of this communication is to point out that a naive application of symplectic integration schemes for hamiltonian systems with constraints such as shake or rattle which preserve holonomic constraints encounters difficulties when applied to the numerical treatment of the equations of general relativity

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

hydrodynamics and time correlation functions for cellular automata hydrodynamic excitations in lattice gas cellular automata are described in terms of equilibrium time correlation functions for the local conserved variables for large space and time scales the linearized hydrodynamic equations are obtained to navier stokes order exact expressions for the associated susceptibilities and transport coefficients are identified in terms of correlation functions the general form of the time correlation functions for conserved densities in the hydrodynamic limit is given and illustrated by some examples suitable for comparison with computer simulation the transport coefficients are related to time correlation functions for the conserved fluxes in a way analogous to the green kubo expressions for continuous fluids the general results are applied for a one component fluid and several types of binary diffusion also discussed are the effects of unphysical slow modes such as staggered particle or momentum densities

volume preserving algorithms for source free dynamical systems in this paper we first expound why the volume preserving algorithms are proper for numerically solving source free systems and then prove all the conventional methods are not volume preserving secondly we give a general method of constructing volume preserving difference schemes for source free systems on the basis of decomposing a source free vector field as a finite sum of essentially 2 dimensional hamiltonian fields and of composing the corresponding essentially symplectic schemes into a volume preserving one lastly we make some special discussions for so called separable source free systems for which arbitrarily high order explicit revertible volume preserving schemes can be constructed

collective diffusion in a lattice gas automaton we report on non mean field and ring kinetic theory calculations of both the momentum autocorrelation function and the collective diffusion coefficient in a diffusive lattice gas automaton for both quantities the ring approximation is calculated exactly a saddle point method yields a leadingt 2 and a subleadingt 5 2 long time tail in the momentum autocorrelation function the ring kinetic corrections to the mean field value of the diffusion coefficient are in good agreement with computer simulations

integration of perturbed initial value problems through reduction theory we propose a method to integrate an initial value problem formed by the differential equations of a perturbed dynamical system plus the initial condition the approach consists of several steps first of all if the original system does not enjoy a continuous symmetry it is transformed up to a certain order of approximation m into an equivalent but equivariant one after truncation at order m second this new symmetry allows us to define a reduction map which produces in its whole phase space a splitting of the transformed equations specifically the latter system is decoupled into two subsystems also initial value problems one of them defined in the orbit space associated with the reduction and the other linear in its coordinates as a third step the reduced subsystems are analytically resolved or numerically integrated using standard methods finally the solution of the original system is recovered by inverting back the reduction map and thereafter the analytic transformations we apply our procedure to three examples

semi lagrangian multistep exponential integrators for index 2 differential algebraic systems implicit explicit imex multistep methods are very useful for the time discretization of convection diffusion pde problems such as the burgers equations and the incompressible navier stokes equations in the latter as well as in pde models of plasma physics and of electromechanical systems semi discretization in space gives rise to differential algebraic dae system of equations often of index higher than 1 in this paper we propose a new class of exponential integrators for index 2 daes arising from the semi discretization of pdes with a dominating and typically nonlinear convection term this class of problems includes the incompressible navier stokes equations the integration methods are based on the backward differentiation formulae bdf and they can be applied without modifications in the semi lagrangian integration of convection diffusion problems the approach gives improved performance at low viscosity regimes

on the backlund transformation and hamiltonian properties of superevaluation equations backlund and darboux backlund transformations are deduced for the superevaluation equations recently deduced by kupershmidt b kuperschmidt a super k dv equation an integrable system preprint utsi tullahoma 1984 phys lett a 102 213 1983 j phys a 17 l 863 1984 and gurses h gurses and o oguz phys lett a 108 437 1985 by a extension of the technique of bpt m boiti f pempinelli and g z tu nuovo cimento b 79 231 1984 to anticommuting variables the bi hamiltonian structure and hence the form of the recursion operator for the lie backlund symmetry for such equations are deduced incidentally some explicit forms of the lie backlund symmetry are also deduced

symplectic numerical integrators in constrained hamiltonian systems recent work reported in the literature suggests that for the long time integration of hamiltonian dynamical systems one should use methods that preserve the symplectic or canonical structure of the flow here we investigate the symplecticness of numerical integrators for constrained dynamics such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies this leads to a constrained hamiltonian system of smaller dimension previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space here it is shown that the very popular verlet method with shake type constraints is equivalent to the same method with rattle type constraints and that the latter is symplectic and time reversible this assumes that the iteration is carried to convergence we also demonstrate the global convergence of the verlet scheme in the presence of shake type and rattle type constraints we perform numerical experiments to compare these methods with the second order backward differentiation method commonly recommended for ordinary differential equations with constraints

new finite dimensional integrable systems and explicit solutions of hirota satsuma coupled kortweg de vries equation the nonlinearization approach is extended to the hirota satsuma coupled kortweg de vries kdv equation associated with a 4 4 matrix spectral problem from which two new finite dimensional integrable hamiltonian systems are obtained in the liouville sense it is shown that the solutions of this coupled kdv equation are reduced to solving a compatible system of ordinary differential equations the reductions of the two hamiltonian systems and their separability are discussed an interesting connection between the two reduced hamiltonian systems in the case of one parameter and known two dimensional integrable systems is revealed as application some explicit solutions of the hirota satsuma coupled kdv equation are derived

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

heterogeneous multiscale methods for mechanical systems with vibrations we are concerned with heterogeneous multiscale methods for the numerical integration of the equations of motion of mechanical systems subjected to fast vibrations we suggest easily formulated asynchronous algorithms that bypass the need for explicitly determining the relations between macro and microstates the new algorithms have good geometric properties and in some cases may incorporate a simplified filtering technique that leads to large savings in computational effort the problems considered may be modeled either by ordinary differential equations state space description or by differential algebraic equations descriptor form a shake shake and a rattle shake algorithm are presented but higher order versions exist

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

application of the symplectic finite difference time domain scheme to electromagnetic simulation an explicit fourth order finite difference time domain fdtd scheme using the symplectic integrator is applied to electromagnetic simulation a feasible numerical implementation of the symplectic fdtd sfdtd scheme is specified in particular new strategies for the air dielectric interface treatment and the near to far field nff transformation are presented by using the sfdtd scheme both the radiation and the scattering of three dimensional objects are computed furthermore the energy conserving characteristic hold for the sfdtd scheme is verified under long term simulation numerical results suggest that the sfdtd scheme is more efficient than the traditional fdtd method and other high order methods and can save computational resources

rattle a velocity version of the shake algorithm for molecular dynamics calculations an algorithm called rattle for integrating the equations of motion in molecular dynamics calculations for molecular models with internal constraints is presented the algorithm is similar to shake which is one of the standard methods for performing such calculations rattle calculates the positions and velocities at the next time from the positions and velocities at the present time step without requiring information about the earlier history like shake it is based on the verlet algorithm and retains the simplicity of using cartesian coordinates for each of the atoms to describe the configuration of a molecule with internal constraints rattle guarantees that the coordinates and velocities of the atoms in a molecule satisfy the internal constraints at each time step rattle has two advantages over shake on computers of fixed precision it is of higher precision than shake since it deals directly with the velocities it is easier to modify rattle for use with the recently developed constant temperature and constant pressure molecular dynamics methods and with the nonequilibrium molecular dynamics methods that make use of rescaling of the atomic velocities

structure preservation for constrained dynamics with super partitioned additive runge kutta methods a broad class of partitioned differential equations with possible algebraic constraints is considered including hamiltonian and mechanical systems with holonomic constraints for mechanical systems a formulation eliminating the coriolis forces and closely related to the euler lagrange equations is presented a new class of integrators is defined the super partitioned additive runge kutta spark methods this class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms a linear stability and convergence analysis of these methods is given spark methods allowing the direct preservation of certain properties are characterized different structures and invariants are considered the manifold of constraints symplecticness reversibility contractivity dilatation energy momentum and quadratic invariants with respect to linear stability and structure preservation the class of s stage lobatto iiia b c c spark methods is of special interest controllable numerical damping can be introduced by the use of additional parameters some issues related to the implementation of a reversible variable stepsize strategy are discussed

reversible methods of runge kutta type for index 2 daes a new interpretation of runge kutta methods for differential algebraic equations daes of index 2 is presented where a step of the method is described in terms of a smooth map smooth also with respect to the stepsize this leads to a better understanding of the convergence behavior of runge kutta methods that are not stiffly accurate in particular our new framework allows for the unified study of two order improving techniques for symmetric runge kutta methods namely post projection and symmetric projection specially suited for solving reversible index 2 daes

explicit exact solutions of some nonlinear evolution equations with their geometric interpretations in this paper the simplest equation method is applied to obtain multiple explicit exact solutions of the combined dispersion equation the hirota satsuma korteweg de vries system and the generalized burgers huxley equation these solutions are constructed on the basis of solutions of bernoulli equation which is used as simplest equation it is shown that this method is very powerful tool for obtaining exact solutions of a large class of nonlinear partial differential equations the geometric interpretation for some of these solutions are introduced

specialized partitioned additive runge kutta methods for systems of overdetermined daes with holonomic constraints we consider a general class of systems of overdetermined differential algebraic equations odaes we are particularly interested in extending the application of the symplectic gauss methods to hamiltonian and lagrangian systems with holonomic constraints for the numerical approximation to the solution to these odaes we present specialized partitioned additive runge kutta spark methods and in particular the new class of s s gauss lobatto spark methods these methods not only preserve the constraints symmetry symplecticness of the flow and variational nature of the trajectories of holonomically constrained hamiltonian and lagrangian systems but they also have an optimal order of convergence 2s a corrected version of this article has been appended at the end of the pdf file

a hamiltonian particle method for non linear elastodynamics particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles they have a great degree of flexibility for instance in dealing with complex large deformations or the fragmentation of solids in this paper a particle method for non linear elastodynamics of compressible and incompressible materials is developed based on a discretization of the lagrangian from which the governing equations of elastodynamics are derived using the principle of least action the discretized lagrangian leads to a finite dimensional hamiltonian system via the legendre transformation if the material is incompressible the hamiltonian system is accompanied by holonomic constraints depending on whether the material is compressible or incompressible the symplectic scheme adopted for numerical time integration is either the stormer verlet scheme or the rattle method respectively the resulting particle method inherits the symplectic structure possessed by the governing equations of elastodynamics in the case of incompressible materials incompressibility is strictly enforced at each time step some numerical tests indicate the excellence of the method for conservation of mechanical energy besides that of linear and angular momenta

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

on the complete solution of the hirota satsuma system through the dressing operator technique the authors have applied the dressing operator method of zakharov et al for obtaining the complete solution of the hirota satsuma coupled system this method helps one essentially to solve the inverse problem associated with a fourth order differential operator also the method of the dressing operator has the distinct advantage of generating solutions other than soliton by varying the boundary condition in an appropriate fashion

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions