higher order approximation methods for the boltzmann equation a higher order time differencing method for the spatially nonhomogeneous boltzmann equation is derived from the integral form of the equation along its characteristic line similar to the splitting method which solves the collisionless equation in the convection step and the spatially homogeneous boltzmann equation in the collision step the present method consists of two steps one of which is the same as the convection step in the splitting method the difference from the splitting method is in the other step where not only the collision term but also its variation along the characteristic line is taken into account correctly the truncation error of this method per time step tiso t3 and its higher order accuracy is demonstrated numerically in the shock propagation problem using the bgk model equation it is shown that such accuracy is never realized in the framework of the conventional splitting formulation which is contrary to bogomolov s result u s s r comput math math phys 28 79 1988 the other higher order methods based on the integral form are also presented furthermore the extension to the stochastic approach modification of the conventional direct simulation monte carlo procedure is proposed

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

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

on the difference equation yn 1 α βe yn γ yn 1 in this paper we investigate the convergence the boundedness and the periodic character of the positive solutions of the difference equationy n email protected email protected y email protected y n 1 n 0 1 2 where the parameters a b c are positive numbers and the initial conditions y 1 y 0 are arbitrary non negative numbers

analysis and implementation of a new constant acceleration subcycling algorithm an algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps with integer time step ratios the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle the algorithm is shown to have good accuracy and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms the implementation of a generalised form of the algorithm with non integer time step ratios is presented c 1997 by john wiley sons ltd

multi time step and two scale domain decomposition method for non linear structural dynamics in this paper we propose a method to improve the means of taking into account the specific time scale and space scale characteristics in time dependent non linear problems this method enables the use of arbitrary time steps in each subdomain these can be coupled by prescribing continuous velocities at the interfaces which are modelled using a dual schur formulation for certain subdomains in space we adopt a two scale resolution technique inspired by the multigrid methods in order to obtain the part of the solution related to small variation lengths on a refined scale and the part corresponding to large variation lengths on a coarse scale for non linear problems we propose an algorithm with a single iteration level to deal with both the non linear equilibrium and the two space scales thanks to a two grid method in which the relaxation steps are performed using a non linear preconditioned conjugate gradient algorithm finally we present an example which demonstrates the feasibility of the method copyright 2003 john wiley sons ltd

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

existence uniqueness and attractivity of prime period two solution for a difference equation of exponential form abstract in this paper we study the existence uniqueness and attractivity of prime period two solution of the difference equation x n 1 a bx n 1 e x n where a b are positive constants and the initial values x 1 x 0 are positive numbers

asymptotic behavior of the positive solutions of an exponential type system of difference equations in this paper we study the existence of a unique positive equilibrium the boundedness persistence and global attractivity of the positive solutions of a system of the following two difference equations x n 1 ax n by n 1 e x n y n 1 cy n dx n 1 e y n n 0 1 where a b c d are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers

a patch that imparts unconditional stability to explicit integrators for langevin like equations this paper extends the results in 8 to stochastic differential equations sdes arising in molecular dynamics it implements a patch to explicit integrators that consists of a metropolis hastings step the patched integrator preserves the sde s equilibrium distribution and is accurate on finite time intervals as a corollary this paper proves the integrator s accuracy in estimating finite time dynamics along an infinitely long solution a first in molecular dynamics the paper also covers multiple time steps holonomic constraints and scalability finally the paper provides numerical tests supporting the theory

study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form in this paper we study the boundedness the persistence and the asymptotic behavior of the positive solutions of the following systems of two difference equations of exponential form xn 1 a be yn c yn 1 y n 1 d e xn f xn 1

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

on dynamic behavior of second order exponential type fuzzy difference equation abstract this paper is concerned with dynamical behaviors of a second order exponential type fuzzy difference equation x n 1 a b e x n c x n 1 n 0 1 where a b c and the initial values x 1 x 0 are positive fuzzy numbers applying generalization of division g division of fuzzy numbers we study the existence of positive fuzzy solution and the global asymptotic behavior of the model moreover two simulation examples are given to show the validity of the theoretical results

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

a study of the stability of subcycling algorithms in structural dynamics algorithms for explicit integration of structural dynamics problems with multiple time steps subcycling are investigated only one such algorithm due to smolinski and sleith has proved to be stable in a classical sense a simplified version of this algorithm that retains its stability is presented however as with the original version it can be shown to sacrifice accuracy to achieve stability another algorithm in use is shown to be only statistically stable in that a probability of stability can be assigned if appropriate time step limits are observed this probability improves rapidly with the number of degrees of freedom in a finite element model the stability problems are shown to be a property of the central difference method itself which is modified to give the subcycling algorithm a related problem is shown to arise when a constraint equation in time is introduced into a time continuous space time finite element model

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

asynchronous multi domain variational integrators for nonlinear hyperelastic solids we present the asynchronous multi domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary value problem in hyperelasticity variational time integration schemes based on the principle of minimal action within the lagrangian framework are constructed for the equation of motion and implemented into a variational finite element framework which is systematically derived from the three field de veubeke hu washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly incompressible materials for efficient parallel computing we use the dual domain decomposition method with local lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains the method for time discretization and the multi domain spatial decomposition enable us to use different types of integrators explicit vs implicit and different time steps on different parts of a computational domain and thus efficiently capture the underlying physics with less computational effort the energy conservation of our nonlinear midpoint asynchronous integration scheme is investigated using the energy method and both local and the global energy error estimates are derived we illustrate the performance of proposed variational multi domain time integrators by means of three examples first the method of manufactured solutions is used to examine the consistency of the formulation in the second example we investigate energy conservation and stability finally we apply the method to the motion of a heterogeneous plane domain where different integrators and time discretization steps are used accordingly with disparate material data of individual parts

a partial velocity approach to subcycling structural dynamics subcycling or the use of different timesteps at different nodes can be an effective way of improving the computational efficiency of explicit transient dynamic structural solutions the method that has been most widely adopted uses a nodal partition extending the central difference method in which small timestep updates are performed interpolating on the displacement at neighbouring large timestep nodes this approach leads to narrow bands of unstable timesteps or statistical stability it also can be in error due to lack of momentum conservation on the timestep interface the author has previously proposed energy conserving algorithms that avoid the first problem of statistical stability however these sacrifice accuracy to achieve stability an approach to conserve momentum on an element interface by adding partial velocities is considered here applied to extend the central difference method this approach is simple and has accuracy advantages the method can be programmed by summing impulses of internal forces evaluated using local element timesteps in order to predict a velocity change at a node however it is still only statistically stable so an adaptive timestep size is needed to monitor accuracy and to be adjusted if necessary by replacing the central difference method with the explicit generalized alpha method it is possible to gain stability by dissipating the high frequency response that leads to stability problems however coding the algorithm is less elegant as the response depends on previous partial accelerations extension to implicit integration is shown to be impractical due to the neglect of remote effects of internal forces acting across a timestep interface c 2002 elsevier science b v all rights reserved

on the difference equation xn 1 a αxn αxn 1 αxn k 2 xn k 1 in this paper we establish invariant relations boundedness persistence strictly oscillatory and semi cycles for the positive solutions of the difference equationx n email protected x n x n 1 x n k 2 x n k 1 n k 1 k email protected 0 email protected 0 email protected 1 2

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

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

stability of the recursive sequence xn 1 α βxn γ xn 1 abstract in this paper we investigate the global asymptotic stability of the recursive sequence xn 1 xn xn 1 n 0 1 where 0 we show that the unique positive equilibrium point of the equation is a global attractor with a basin that depends on the conditions posed on the coefficients

analysis of the neumann problem along a straight edge we show there exists an l p solution for p 2 to the open image in new window neumann problem on an edge domain in open image in new window 2 for 0 1 forms and we explicitly compute the singularities which are of complex logarithmic and arctangent type along the edge of the solution

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

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