a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

the midpoint scheme and variants for hamiltonian systems advantages and pitfalls the implicit midpoint scheme like higher order gauss collocation schemes is algebraically stable and symplectic and it preserves quadratic integral invariants it may appear particularly suitable for the numerical solution of highly oscillatory hamiltonian systems such as those arising in molecular dynamics or structural mechanics because there is no stability restriction when it is applied to a simple harmonic oscillator although it is well known that the midpoint scheme may also exhibit instabilities in various stiff situations one might still hope for good results when resonance type instabilities are avoided in this paper we investigate the suitability of the midpoint scheme for highly oscillatory frictionless mechanical systems where the step size k is much larger than the system s small parameter varepsilon in the case that the solution remains bounded as varepsilon rightarrow 0 we show that in general one must require that k 2 varepsilon be small enough or else even the errors in slowly varying quantities like the energy may grow undesirably especially when fast and slow modes are tightly coupled or worse the computation may yield misleading information in some cases this may already happen when k o varepsilon the same holds for higher order collocation at gaussian points the encountered restrictions on k are typically still better than the corresponding ones for explicit schemes

a family of symplectic integrators stability accuracy and molecular dynamics applications the following integration methods for special second order ordinary differential equations are studied leapfrog implicit midpoint trapezoid stormer verlet and cowell numerov we show that all are members or equivalent to members of a one parameter family of schemes some methods have more than one common form and we discuss a systematic enumeration of these forms we also present a stability and accuracy analysis based on the idea of modified equations and a proof of symplecticness it follows that cowell numerov and lim2 a method proposed by zhang and schlick are symplectic a different interpretation of the values used by these integrators leads to higher accuracy and better energy conservation hence we suggest that the straightforward analysis of energy conservation is misleading

stability of asynchronous variational integrators the adoption of multiple time step integrators can provide substantial computational savings for mechanical systems with multiple time scales however the scope of these savings may be limited by the range of allowable time step choices in this paper we analyze the linear stability of the fully asynchronous methods termed avi for asynchronous variational integrators we perform a detailed analysis for the case of a one dimensional particle moving under the action of a soft and a stiff quadratic potential integrated with two time steps in rational ratios in this case we provide sufficient conditions for the stability of the method these generalize to the fully asynchronous avi case the results obtained for synchronous multiple time stepping schemes such as r respa which show resonances when the larger time step is a multiple of the effective half period of the stiff potential additionally we numerically investigate the appearance of instabilities based on the experimental observations we conjecture the existence of a dense set of unstable time steps when arbitrary rational ratios of time steps are considered in this way unstable schemes for arbitrarily small time steps can be obtained however the vast majority of these instabilities are extremely weak and do not present an obstacle to the use of these integrators we then applied these results to analyze the stability of multiple time step integrators in the more complex mechanical systems arising in molecular dynamics and solid dynamics we explained why strong resonances are ubiquitously found in the former while rarely encountered in the latter finally in this paper we introduce a formulation of avi that highlights the symplectic nature of the algorithm complementing those introduced earlier by other authors

chaotic principle an experimental test abstract the chaotic hypothesis discussed in gallavotti and cohen 1995 is tested experimentally in a simple conduction model besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which as the forcing strength grows the attractor ceases to be an anosov system and becomes an axiom a attractor a first text of new predictions is also attempted

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

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

molecular dynamics smoothed particle applied mechanics and irreversibility abstract we explore the relationship of monaghan s version of smoothed particle hydrodynamics here called smoothed particle applied mechanics to nonequilibrium molecular dynamics we first use smoothed particles to model the simplest possible linear transport problems as well as a liquid drop problem we then consider both gas phase and dense fluid versions of rayleigh benard convection all in two space dimensions we also discuss the possibility of combining the microscopic and macroscopic techniques in a hybrid scheme well suited to the massively parallel modelling of large scale nonequilibrium flows

structure of characteristic lyapunov vectors in anharmonic hamiltonian lattices in this work we perform a detailed study of the scaling properties of lyapunov vectors lvs for two different one dimensional hamiltonian lattices the fermi pasta ulam and 4 models in this case characteristic also called covariant lvs exhibit qualitative similarities with those of dissipative lattices but the scaling exponents are different and seemingly nonuniversal in contrast backward lvs obtained via gram schmidt orthonormalizations present approximately the same scaling exponent in all cases suggesting it is an artificial exponent produced by the imposed orthogonality of these vectors we are able to compute characteristic lvs in large systems thanks to a bit reversible algorithm which completely obviates computer memory limitations

verlet i r respa impulse is limited by nonlinear instabilities this paper shows that in molecular dynamics md when constant energy nve simulations of newton s equations of motion are attempted using the multiple time stepping mts integrator verlet i r respa impulse there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period s of the fastest motion s in the system this is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem the numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein but also the constrained dynamics simulation of a solvated protein representing a range of simulation protocols commonly in use by biomolecular modelers the observed and predicted instabilities match exactly previous work has identified and explained a linear instability for verlet i r respa impulse at around half the period of the fastest motion mandziuk and schlick discovered nonlinear resonances in single time stepping md integrators but unstable nonlinear resonances for mts integrators are reported here for the first time this paper also offers an explanation on the instability of mts constrained molecular dynamics simulations of explicitly solvated proteins more aggressive multiple step sizes are possible with mild langevin coupling or targeted langevin coupling and its combination with the mollified impulse method permits step sizes 3 to 4 times larger thanverlet i r respa impulse while still retaining some accuracy

multidimensional activated rate processes with slowly relaxing mode abstract the multidimensional kramers problem is considered in the case of highly anisotropic friction as a development of our previous results involving the study of several new regimes of activated rate process we construct a theory which incorporates all these regimes and treats the process in every case in a uniform manner to do this we take advantage of the friction anisotropy and reduce the initial multidimensional fokker planck equation to a set of two effective one dimensional equations these equations describe diffusive motion along the slow coordinate in two effective potential wells and inter well transitions due to the fast coordinate motion to calculate the rate constant on the basis of the set we use an original method which is a generalization of the kramers method as a result we obtain a new formula for the rate constant which depending on the friction anisotropy level is smoothly changed from the well known kramers langer formula to new ones obtained in our previous papers

time reversible continuum mechanics levesque and verlet developed a time reversible and bit reversible computational leapfrog algorithm their algorithm uses integer arithmetic and is exactly time reversible to the last computational bit describing the particle coordinates we generalize their idea developed for atomistic molecular dynamics to smoothed particle continuum mechanics in the special case of a two dimensional isentropic ideal gas these two approaches one microscopic and the other macroscopic are isomorphic in the more general nonadiabatic case but still without dissipative terms our continuum extension of the leapfrog scheme remains stable and also exhibits the exact time and bit reversibility associated with levesque and verlet s atomistic approach

asymptotics of repeated interaction quantum systems abstract a quantum system s interacts in a successive way with elements e of a chain of identical independent quantum subsystems each interaction lasts for a duration and is governed by a fixed coupling between s and e we show that the system initially in any state close to a reference state approaches a repeated interaction asymptotic state in the limit of large times this state is periodic in time and does not depend on the initial state if the reference state is chosen so that s and e are individually in equilibrium at positive temperatures then the repeated interaction asymptotic state satisfies an average second law of thermodynamics

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

random repeated quantum interactions and random invariant states we consider a generalized model of repeated quantum interactions where a system mathcal h is interacting in a random way with a sequence of independent quantum systems mathcal k n n geq 1 two types of randomness are studied in detail one is provided by considering haar distributed unitaries to describe each interaction between mathcal h and mathcal k n the other involves random quantum states describing each copy mathcal k n in the limit of a large number of interactions we present convergence results for the asymptotic state of mathcal h this is achieved by studying spectral properties of random quantum channels which guarantee the existence of unique invariant states finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble

molecular dynamics and time reversibility we present a time symmetrical integer arithmetic algorithm for numerical molecular dynamics simulations of classical fluids this algorithm is used to illustrate through concrete examples that time asymmetric evolutions are typical for systems of many particles evolving according to reversible microscopic dynamics and to calculate the asymptotic behavior of the velocity autocorrelation function with an improved accuracy the equivalence between equilibrium time averages and microcanonical ensemble averages is checked via two new sampling methods for computing microcanonical averages of classical systems

parallel asynchronous variational integrators this paper presents a scalable parallel variational time integration algorithm for nonlinear elastodynamics with the distinguishing feature of allowing each element in the mesh to have a possibly different time step furthermore the algorithm is obtained from a discrete variational principle and hence it is termed parallel asynchronous variational integrator pavi the underlying variational structure grants it outstanding conservation properties based on a domain decomposition strategy pavi combines a careful scheduling of computations with fully asynchronous communications to provide a very efficient methodology for finite element models with even mild distributions of time step sizes numerical tests are shown to illustrate pavi s performance on both slow and fast networks showing scalability properties similar to the best parallel explicit synchronous algorithms with lower execution time finally a numerical example in which pavi needs 100 times less computing than an explicit synchronous algorithm is shown copyright 2006 john wiley sons ltd

comments on some aspects of boltzmann h theorem using reversible molecular dynamics we calculate the boltzmannh function for a repulsive lennard jones fluid using levesque and verlet s reversible molecular dynamics algorithm we find as predicted by jaynes that for certain initial states h increases as a function of time we also discuss the grad limit within such a simulation

symplectic scheme and the poynting vector in reverse time migration abstractwe developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method rem this solution can be used for seismic modeling and reverse time migration rtm in seismic modeling and rtm spatial derivatives are usually calculated by finite differences fds or by the fourier method and the time evolution is normally obtained by a second order fd approach if the spatial derivatives are computed by higher order fd schemes then the time step needs to be small enough to avoid numerical dispersion therefore increasing the computational time however by using rem with the fourier method for the spatial derivatives we can apply the proposed method to propagate the wavefield for larger time steps moreover if the appropriate number of expansion terms is chosen this method is unconditionally stable and propagates seismic waves free of numerical dispersion the use of a symplectic numerical scheme provides the solution of the wave equation an

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

explicit symplectic integrators for solving nonseparable hamiltonians by exploiting the error functions of explicit symplectic integrators for solving separable hamiltonians i show that it is possible to develop explicit time reversible symplectic integrators for solving nonseparable hamiltonians of the product form the algorithms are unusual in that they are of fractional orders

dft modal analysis of spectral element methods for the 2d elastic wave equation the dft modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete fourier transform domain sampled in the mesh nodes this technique provides a natural matching of exact and approximate modes of propagation we extend this technique to spectral element methods for the 2d isotropic elastic wave equation by using a rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation taking full advantage of the tensor product representation of the spectral element matrices numerical experiments illustrate the dependence of dispersion errors on the grid resolution polynomial degree and discretization in time we consider spectral element methods with chebyshev and legendre collocation points

symmetric explicit numerical integrator for molecular dynamics equations of motion with a generalized friction a general mathematical scheme to construct symmetric explicit numerical integrators of newtonian equations of motion endowed with a generalized friction is provided for a molecular dynamics md study the exact integrations are done for all the decomposed vector fields including the one that contains the friction term on the basis of the symmetric composition scheme with the adjoint for the resulting maps integrators with any local order of accuracy can be systematically constructed among them the second order p2s1 integrator gives the least evaluation of atomic force and potential which are most time consuming in md simulations as examples of the friction function three functional types are considered constant laurent polynomial and exponential with respect to the kinetic energy several md equations of motion fall into these categories and the numerical examinations of their integrators using model systems give positive results on the accuracy and efficiency the extended phase space scheme also presents an invariant function which allows us to easily detect numerical errors in the integration process by monitoring the function value

an energy momentum conserving algorithm using the variational formulation of visco plastic updates in this paper we use the variational formulation of elasto plastic updates proposed by ortiz and stainier comput methods appl mech eng 1999 171 419 444 in the context of consistent time integration schemes we show that such a formulation is well suited to obtain a general expression of energy momentum conserving algorithms moreover we present numerical examples that illustrate the efficiency of our developments copyright 2005 john wiley sons ltd

a singular perturbation approach to kramers diffusion problem we consider kramers diffusion problem which seeks to calculate the rate of escape of a particle from one potential well over a barrier to another presumably deeper and therefore more stable well though kramers introduced the problem as a model for chemical reactions it applies to numerous rate processes including atomic migration and ionic conductivity in crystals and transitions due to noise between stable states of dynamical systems with multi stable states to name but a few we propose a new approach not based on a fokker planck equation but rather on the solution of a singularly perturbed boundary value problem specifically we relate the rate of escape to the first passage time from the domain of attraction of the stable point corresponding to the first well the first passage time is then characterized via the ito calculus as a solution of an elliptic partial differential equation of singular perturbation type finally this equation is solved asymptotically by methods previously develop

assumed strain finite element methods for conserving temporal integrations in non linear solid dynamics this paper presents a new class of assumed strain finite elements to use in combination with general energy momentum conserving time stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time the case of interest corresponds to nearly incompressible material responses in the fully non linear finite strain elastic and elastoplastic ranges the new elements consider the classical scaling of the deformation gradient with an assumed jacobian its determinant defined locally through a weighted averaging procedure at the element level the key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or b bar operator the developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta and the conservation dissipation law of energy for elastic and inelastic problems respectively care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration with these developments at hand a new general b bar operator is introduced that satisfies these conditions the new operator not only accounts for the spatial interpolations e g bilinear displacements with piece wise constant volume but also depends on the discrete structure of the equations in time the aforementioned conservation dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes unconditionally of the time step size and the material model elastic or elastoplastic different finite elements are considered in this framework including quadrilateral and triangular elements for plane problems and brick elements for three dimensional problems several representative numerical simulations are presented involving in particular the use of energy dissipating momentum conserving time stepping schemes recently developed by the author and co workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements including these conservation dissipation properties in time and their locking free response in the quasi incompressible case copyright 2007 john wiley sons ltd

repeated interaction quantum systems van hove limits and asymptotic states we establish the existence of two weak coupling regime effective dynamics for an open quantum system of repeated interactions vanishing strength and individual interaction duration respectively this generalizes known results attal and joye in j stat phys 126 1241 1283 2007 in that the von neumann algebras describing the system and the chain element may not be of finite type then but now assuming that the small system is of finite type we prove that both effective dynamics capture the long term behavior of the system existence of a unique asymptotic state for them implies the same property for the respective exact dynamics provided that the perturbation parameter is sufficiently small the zero th order term in a power series expansion in the perturbation parameter of such an asymptotic state is given by the asymptotic state of the effective dynamics we conclude by working out the case in which the small system and the chain element are spins

random repeated interaction quantum systems we consider a quantum system mathcal s interacting sequentially with independent systems mathcal e m m 1 2 before interacting each mathcal e m is in a possibly random state and each interaction is characterized by an interaction time and an interaction operator both possibly random we prove that any initial state converges to an asymptotic state almost surely in the ergodic mean provided the couplings satisfy a mild effectiveness condition we analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics

microscopic origins of irreversible macroscopic behavior time asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems it can be understood as arising naturally from time symmetric microscopic laws in accord with the ideas of thompson maxwell and boltzmann alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles probability distributions of certain systems having only a few degrees of freedom are in my opinion unnecessary misguided and misleading

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

error analysis of symplectic multiple time stepping most time dependent problems obtain only limited benefit from variable stepsize and to exploit more fully the existence of multiple time scales it is necessary to use a multiplicity of stepsizes simultaneously for example many systems of odes have right hand sides which naturally decompose into a sum of terms with different time scales in which case the various terms can be sampled with different stepsizes an idea called multiple time stepping mts addressed here is the question of how to choose these stepsizes in the context of molecular dynamics this application and others are often modeled by hamiltonian systems for which symplectic integrators are often preferred because they introduce discretization error that can be very nearly interpreted as a perturbation to the scalar hamiltonian function itself the method analyzed here is a symplectic mts method that has received recent attention analyzed is a problem that models the bonded interactions in molecular dynamics md the method of analysis is quite novel in that it utilizes a slight change of variables that is more favorable to the numerical solution without seriously affecting the quantities of interest the study concludes with a specific prescription for determining stepsize ratios

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

time reversibility in nonequilibrium thermomechanics abstract time reversibility of dynamical systems has had many different interpretations for the governing differential equations for the numerical approximations to their solutions for the maps which caricature these equations and their solutions and for the phase space distributions describing ensembles of systems here i discuss some of these views taken from recent research literature and relate them to my own experience which is primarily computational i emphasize the usefulness of thermomechanics a simple generalization of conventional molecular dynamics which leads directly inexorably and paradoxically from hamilton s principle of least action to the multifractal time irreversible phase space behavior summed up by the macroscopic second law of thermodynamics

generalization of the kramers langer theory decay of the metastable state in the case of strongly anisotropic friction the problem of escape of a classical particle from a multidimensional potential well due to the influence of random forces is studied a new solution to the problem is obtained which shows that with fairly large friction anisotropy the qualitative picture of the process principally differs from that underlying the conventional multidimensional kramers langer theory according to the new process picture particles escape the well before reaching the saddle point the limiting stage of the process is the particle attainment of the transition region i e well dynamics not during passage through the barrier regions as the conventional theory suggests the new expression for the rate constant predicts a slower process rate in comparison with the kramers langer theory with decreasing friction anisotropy the new expression coincides with the conventional one

a family of noniterative integration methods with desired numerical dissipation a new family of unconditionally stable integration methods for structural dynamics has been developed which possesses the favorable numerical dissipation properties that can be continuously controlled in particular it can have zero damping this numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes whereas the low frequency modes are almost unaffected the most important improvement of this family method is that it involves no nonlinear iterations for each time step and thus it is very computationally efficient when compared with a general second order accurate integration method such as the constant average acceleration method copyright 2014 john wiley sons ltd

nonlinear stability analysis of area preserving integrators linear stability analysis is inadequate for integrators designed for nondissipative systems such as hamiltonian systems in which nonlinear effects are often decisive mathematical theory exists kam theory for rigorous analysis of small perturbations from equilibria but it needs to be expressed in a form that is more easily applicable to the study of area preserving maps we have pursued this obtaining a completely rigorous nonlinear stability analysis for elliptic equilibria based on the moser twist theorem and a result of cabral and meyer nonlinearity 12 1999 pp 1351 1362 together with the theory of normal forms for hamiltonian systems the result is a determination of necessary and sufficient conditions for stability these conditions are sharpened for the case of reversible maps and applied to the symplectic members of the newmark family of integrators which includes the leapfrog the implicit midpoint and the stormer cowell methods nonlinear stability limits are more severe than those of linear theory as an example the leapfrog scheme actually has a step size limitation of 71 of that predicted by linear analysis

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

explicit symplectic integrators using hessian vector products in 1991 rowlands proposed an effectively fourth order effectively two stage explicit symplectic integrator based on using a hessian vector product to modify the force evaluation in the leapfrog method and evidence indicates that for modest accuracy this method is highly competitive here we explore the possible existence of even more efficient fourth order explicit symplectic integrators also based on the use of hessian vector products and the concept of effective order first it is shown that the cost of a force evaluation plus a hessian vector product is less than twice the cost of the force alone for a sum of two body interactions then a new method is found that is generally better than both the method of rowlands and that of calvo according to both a theoretical measure of the error and limited numerical experiments the basic motivation behind the new method is quite simple do a hessian vector computation only every other step significantly cutting costs while only marginally increasing the error the idea of effective order means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibility of postprocessing the values obtained by the basic method but at output points only for some applications processing is unnecessary but in any case processing has been shown to be possible at low additional cost the derivation of the new method illustrates how to simplify by the use of h series the determination of parameters for methods of increased effective accuracy