hyperbolic type transport equations abstract in recent years hyperbolic type transport equations have acquired a great deal of importance in problems ranging from theoretical physics to biology in spite of their greater mathematical difficulty as compared with their parabolic type analogs arising from the framework of linear irreversible thermodynamics they have in many ways superseded the latter ones although the use of this type of equations is well known since the last century through the telegraphist equation of electromagnetic theory their use in studying several problems in transport theory is hardly fifty years old in fact the first appearance of a hyperbolic type transport equation for the problem of heat conduction dates back to cattaneos work in 1948 three years later in 1951 s goldstein showed how in the theory of stochastic processes this type of an equation is obtained in the continuous limit of a one dimensional persistent random walk problem after that other phenomenological derivations have been offered for such equations the main purpose of this paper is to critically discuss a derivation of a hyperbolic type fokker planck equation recently presented using the same ideas as m s green did in 1952 to provide the stochastic foundations of irreversible statistical mechanics arguments are given to show that such an equation as well as transport equations derived from it by taking appropriate averages are at most approximate and that a much more detailed analysis is required before asserting their validity

time finite element methods a unified framework for numerical discretizations of odes abstract we present a unified framework for the numerical discretization of odes based on time finite element methods we relate time finite element methods to runge kutta methods with infinitely many stages by means of the corresponding numerical quadrature we establish the relation between time finite element methods and partitioned runge kutta methods we also provide order estimates and superconvergence of the corresponding numerical methods in use of the simplifying assumptions we apply time finite methods to hamiltonian systems and investigate the conservation of energy and symplectic structure for the resulting numerical discretizations for hamiltonian systems we also construct new classes of symplectic integrators by combining different time finite element methods and verify the results by performing some numerical experiments

generalisation du critere de beurling nyman pour l hypothese de riemann nous generalisons dans cet article le critere de beurling nyman qui concerne la fonction de riemann a une large classe de series de dirichlet nous etablissons donc une correspondance entre la densite d un certain sous espace de fonctions dans et la localisation des zeros d une serie de dirichlet nous utilisons pour obtenir ce resultat la structure de l espace de hardy du demi plan abstract we generalise beurling nyman s criterion already known for the riemann function to a larger class of dirichlet series we reveal a link between the density of some subspace of functions in and the localization of the zeros of a dirichlet series to do so we use the structure of the hardy space of the half plan

the continuum limit of a two dimensional persistent random walk abstract we consider a two dimensional persistent random walk in which the motion consists of alternative steps along one of two vectors a and b it is shown that the continuum limit of the evolution equation is not a two but rather a one dimensional telegrapher s equation

random evolutions are driven by the hyperparabolic operators we resolve the long standing problem of describing the multidimensional random evolutions by means of the telegraph equations this problem was posed by mark kac more than 50 years ago and has become the subject of intense discussion among researchers on whether the multidimensional random flights could be described by the telegraph equations similarly to the one dimensional case we give the exhaustive answer to this question and show that the multidimensional random evolutions are driven by the hyperparabolic operators composed of the telegraph operators and their integer powers the only exception is the 2d random flight whose transition density is the fundamental solution to the two dimensional telegraph equation the reason of the exceptionality of the 2d case is explained we also show that under the standard kac s condition the governing hyperparabolic operator turns into the generator of the brownian motion

random motions at finite speed in higher dimensions we present a general method of studying the transport process bold x t t 0 in the euclidean space m m 2 based on the analysis of the integral transforms of its distributions we show that the joint characteristic functions of bold x t are connected with each other by a convolution type recurrent relation this enables us to prove that the characteristic function fourier transform of bold x t in any dimension m 2 satisfies a convolution type volterra integral equation of second kind we give its solution and obtain the characteristic function of bold x t in terms of the multiple convolutions of the kernel of the equation with itself an explicit form of the laplace transform of the characteristic function in any dimension is given the complete solution of the problem of finding the initial conditions for the governing partial differential equations is given

numerical methods for hermitian unitary differential systems in recent years some numerical methods have been developed to integrate matrix differential systems whose solutions are unitary matrices in this paper we study numerical methods for the special class of unitary systems the solutions of which are hermitian matrices throughout the evolution several numerical comparisons with known unitary integrators are reported

reconstructing the transition rate function of a broadwell random walk from exit times in this paper we utilize the layer stripping method a method used in seismology by geophysicists to study a stochastic inverse problem arising from a one dimensional broadwell process the broadwell process can be described as a random walk of a particle that transitions with a spatially dependent flip rate between two states associated with the same speed but with different signs of velocities our goal is to reconstruct the spatially dependent flip rate function of a one dimensional broadwell process from exit time distributions out of a finite interval in principle we are able to reconstruct flip rate functions with error within o 10 2 from o 10 5 exit times when the particle speed is unity for smaller particle speeds the noise increases for a fixed number of exit times this method is less time consuming compared with traditional projection methods which involve optimization

an extended discrete gradient formula for oscillatory hamiltonian systems in this paper incorporating the idea of the discrete gradient method into the extended runge kutta nystrom integrator we derive and analyze an extended discrete gradient formula for the oscillatory hamiltonian system with the hamiltonian where represents generalized positions represents generalized momenta and is a symmetric and positive semi definite matrix the solution of this system is a nonlinear oscillator basically many nonlinear oscillatory mechanical systems with a partitioned hamiltonian function lend themselves to this approach the extended discrete gradient formula presented in this paper exactly preserves the energy h p q we derive some properties of the new formula the convergence is analyzed for the implicit schemes based on the discrete gradient formula and it turns out that the convergence of the implicit schemes based on the extended discrete gradient formula is independent of m which is a significant property for the oscillatory hamiltonian system thus it transpires that a larger step size can be chosen for the new energy preserving schemes than that for the traditional discrete gradient methods when applied to the oscillatory hamiltonian system illustrative examples show the competence and efficiency of the new schemes in comparison with the traditional discrete gradient methods in the scientific literature

random walk with persistence the exact analytic result is obtained for the fourier transform of the generating functionf r s 0 s n p r n wherep r n is the probability density for the end to end distancer inn steps of a random walk with persistence the moments r 2 n r 4 n and r 6 n are calculated and approximate results forp r n and r 1 n are given

explicit volume preserving and symplectic integrators for trigonometric polynomial flows we introduce explicit volume preserving and symplectic integrators for the case of generalized trigonometric polynomial flows the method is demonstrated using the arter flow and computational trials are conducted using a 4 dimensional vector field

integral preserving integrators ordinary differential equations having a first integral may be solved numerically using one of several methods with the integral preserved to machine accuracy one such method is the discrete gradient method it is shown here that the order of the method can be bootstrapped repeatedly to higher orders of accuracy the method is illustrated using the henon heiles system

a stabilization of multistep linearly implicit schemes for dissipative systems we consider numerical integration of dissipative gradient systems for such systems a class of special stable integrators that strictly maintain dissipation is known but they generally yield expensive fully implicit schemes and when the system is large linearization is indispensable for practical efficiency however this can in turn destroy the originally expected stability and so far no effective principle has been formulated for a stable linearization in this note we point out that the behavior of the linearized schemes can be understood from a dynamical systems theory viewpoint and propose a simple principle for a stable linearization

improving the accuracy of the avf method the average vector field avf method is a b series scheme of the second order as a discrete gradient method it preserves exactly the energy integral for any canonical hamiltonian system we present and discuss two locally exact and energy preserving modifications of the avf method avf lex of the third order and avf slex of the fourth order applications to spherically symmetric potentials are given including a compact explicit expression for the avf scheme for the coulomb kepler problem

a lyapunov type theorem for dissipative numerical integrators with adaptive time stepping the asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time stepping can be fully investigated by a lyapunov type theorem on continuous and discrete dynamical systems respectively however once adaptive time stepping is involved such theories cease to work and usually the dynamics should be investigated from the past instead of the standard forward way such as in terms of pullback attractors in this paper we present a different approach we stick to a forward definition of limit sets and show that still we can establish a lyapunov type theorem which reveals the precise asymptotic behavior of adaptive time stepping integrators in the presence of a discrete lyapunov functional

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

some applications of persistent random walks and the telegrapher s equation abstract a persistent random walk can be regarded as a multidimensional markov process the bias free telegraphers equation is 2 p t 2 1 t p t v 2 2 p it can be regarded as interpolating between the wave equation t and the diffusion equation t 0 previously it has found application in thermodynamics cf the review in rev mod phys 61 1989 41 62 1990 375 more recent applications are reviewed in the present article

invariants and numerical methods for odes the preservation of some invariants is important during the numerical integration of odes in some cases failure to maintain certain invariants leads to physically impossible solutions in other cases to instability some authors report that the long term macroscopic characteristics of a solution are better represented if certain invariants are maintained this paper considers two types of invariants point invariants which are functions along a particular trajectory and differential invariants which are invariant relations between neighboring trajectories such as symplectic invariants conventional numerical methods usually introduce o hp 1 errors in an invariant at each numerical step and these will accumulate over the integration interval two approaches to preventing this accumulation are 1 to find methods that maintain the invariants within round off error or 2 to find methods that maintain another invariant which is no more than o hq different from the invariant satisfied by the ode these approaches will be illustrated with some examples the major objective of this paper is to consider the potential of general methods that is methods that do not depend on the specific differential equation to maintain the invariants

zero free regions for dirichlet series in this paper we are interested in explicit zero free discs for some dirichlet series and we also study a general beurling nyman criterion for l functions our results generalize and improve previous results obtained by n nikolski and by a de roton as a concrete application we get for example a beurling nyman type criterion for the siegel zero problem

moments of the chain reaction distribution abstract we show how to find the moments of the continuous random walk with exponential pausing time density in any number of dimensions the equivalence between this model and a lorentz gas is established asymptotic properties are discussed finally we show how these results can be easily extended to the chain reaction case in which the mean number of scattered particles changes after each scattering

two dimensional telegraphic processes and their fractional generalizations we study the planar motion of telegraphic processes we derive the two dimensional telegrapher s equation for isotropic and uniform motions starting from a random walk model which is the two dimensional version of the multistate random walk with a continuum number of states representing the spatial directions we generalize the isotropic model and the telegrapher s equation to include planar fractional motions earlier we worked with the one dimensional version masoliver phys rev e 93 052107 2016 prehbm2470 004510 1103 physreve 93 052107 and derived the three dimensional version masoliver phys rev e 96 022101 2017 prehbm2470 004510 1103 physreve 96 022101 an important lesson is that we cannot obtain the two dimensional version from the three dimensional or the one dimensional one from the two dimensional result each dimension must be approached starting from an appropriate random walk model for that dimension

memory based persistence in a counting random walk process this paper considers a memory based persistent counting random walk based on a markov memory of the last event this persistent model is a different than the weiss persistent random walk model however leading thereby to different results we point out to some preliminary result in particular we provide an explicit expression for the mean and the variance both nonlinear in time of the underlying memory based persistent process and discuss the usefulness to some problems in insurance finance and risk analysis the motivation for the paper arose from the counting of events whether rare or not in insurance that presume that events are time independent and therefore based on the poisson distribution for counting these events

energy preserving numerical schemes of high accuracy for one dimensional hamiltonian systems we present a class of non standard numerical schemes which are modifications of the discrete gradient method they preserve the energy integral exactly up to the round off error the considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order in numerical experiments we compare our integrators with some other numerical schemes including the standard discrete gradient method the leap frog scheme and a symplectic scheme of fourth order we study the error accumulation for a very long time and the conservation of the energy integral

on the equivalence between sor type methods for linear systems and the discrete gradient methods for gradient systems abstract inspired by the iterative nature of many discretization methods for continuous dynamical systems connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems have been studied since 1970s for stationary iterative methods solving linear systems chu 1988 2008 discussed a connection to continuous dynamical systems by using the explicit euler method however further understanding of stationary iterative methods might be limited due to the use of the explicit euler method this paper presents a new connection based on the so called discrete gradient methods between sor type methods and gradient systems there the key of the discussion is the equivalence between sor type methods and the discrete gradient methods applied to gradient systems the discussion leads to new interpretations for sor type methods for example a new derivation of sor type methods is found these methods monotonically decrease a certain quadratic function and a new interpretation of the relaxation parameter is obtained besides while studying the new connection a new discrete gradient is also obtained

scaling relationships for anisotropic random walks aspects of transport in a highly multiple scattering environment are investigated by examining random walkers moving in media having anisotropic angular scattering cross sections turn angle distributions a general expression is obtained for the mean square displacement x2 of a random walker executing ann step walk in an infinite homogeneous material and results are used to predict scaling relations for the probability that a walker returns to the planar surface of a semi infinite medium at a distance from the point of its insertion

evolution of the initial box signal for time fractional diffusion wave equation in a case of different spatial dimensions abstract in the case of time fractional diffusion wave equation considered in the spatial domain x evolution of initial box signal was investigated by mainardi f mainardi fractional relaxation oscillation and fractional diffusion wave phenomena chaos solitons fractals 7 1996 1461 1477 in the present paper we supplement mainardi s results with additional numerical calculations illustrating the behavior of the solution and solve the corresponding problems for axisymmetric and central symmetric cases the obtained results show an unusual behavior of solutions

asymptotic relation for the transition density of the three dimensional markov random flight on small time intervals we consider the markov random flight varvec x t t 0 in the three dimensional euclidean space mathbb r 3 with constant finite speed c 0 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate lambda 0 series representations for the conditional characteristic functions of varvec x t corresponding to two and three changes of direction are obtained based on these results an asymptotic formula as t rightarrow 0 for the unconditional characteristic function of varvec x t is derived by inverting it we obtain an asymptotic relation for the transition density of the process we show that the error in this formula has the order o t 3 and therefore it gives a good approximation on small time intervals whose lengths depend on lambda an asymptotic formula as t rightarrow 0 for the probability of being in a three dimensional ball of radius r ct is also derived estimate of the accuracy of the approximation is analysed

generalized two stream theory for photon migration abstract recent work has indicated that a telegrapher s type equation may accurately describe the diffusion of photons in highly turbid media specially during the ballistic regime of the process on the other hand calculations that have been made using the persistent random walk model prw appear to contradict this result since the telegrapher s equation for the case of very simple lattice geometries does not exist using prw in 2 and 3 dimensions in the present work we show how the continuous diffusion equations in 2d and 3d of prw in the weak coupling limit provide an alternative theory for mesoscopic photon diffusion showing that this problem can be studied using either the traditional telegrapher s equation or by the new approach provided by the continuum and discrete versions of prw

locally exact modifications of numerical schemes we present a new class of exponential integrators for ordinary differential equations locally exact modifications of known numerical schemes local exactness means that they preserve the linearization of the original system at every point in particular locally exact integrators preserve all fixed points and are a stable we apply this approach to popular schemes including euler schemes the implicit midpoint rule and the trapezoidal rule we found locally exact modifications of discrete gradient schemes for symmetric discrete gradients and coordinate increment discrete gradients preserving their main geometric property exact conservation of the energy integral for arbitrary multidimensional hamiltonian systems in canonical coordinates numerical experiments for a two dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium much higher than suggested by the order of new schemes locally exact modification sometimes increases the order but in many cases leaves it unchanged

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

probability law for the euclidean distance between two planar random flights we consider two independent symmetric markov random flights z1 t and z2 t performed by the particles that simultaneously start from the origin of the euclidean plane mathbb r 2 in random directions distributed uniformly on the unit circumference s1 and move with constant finite velocities c1 0 c2 0 respectively the new random directions are taking uniformly on s1 at random time instants that form independent homogeneous poisson flows of rates 1 0 2 0

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

statistics of a filtered telegraph signal equations are derived which describe the time evolution of the probability density and corresponding characteristic function of a telegraph signal which has passed through a detuned lorentzian filter a closed form expression for the characteristic function is obtained for the tuned case and the predicted joint statistics and correlation properties are reviewed in the context of earlier results low order correlation properties for the more general detuned case are calculated it is shown that the stationary single interval statistics can be generated by a random phasor moving in a space of fractional dimensions and that a simple transform of variable leads to distributions which are stable in this space

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

some two and three dimensional persistent random walks abstract we formulate non markovian versions of the persistent random walk in two and three dimensions and in continuous time these models can be regarded as being generalizations of the original pearson random walk and of the freely jointed chain which has been of some importance in polymer physics solutions for the probability density of the displacement of the random walker can be furnished for a restricted essentially markovian set of these models it is shown that in two dimensions the solution to one of these models is equivalent to the solution to an inhomogeneous telegrapher s equation it does not appear to be possible starting from a similar model in three dumensions to find any form of the telegrapher s equation that follows from our solution of the equations in the fourier laplace transform domains

zero free regions for dirichlet series ii in this paper we continue some work devoted to explicit zero free discs for a large class of dirichlet series in a previous article such zero free regions were described using some spaces of functions which were defined with some technical conditions here we give two different natural ways in order to remove those technical conditions in particular this allows to right down explicit zero free regions differently and to obtain for them an easier description useful for direct applications

stochastic processes driven by dichotomous markov noise some exact dynamical results stochastic processes defined by a general langevin equation of motion where the noise is the non gaussian dichotomous markov noise are studied a non fokker planck master differential equation is deduced for the probability density of these processes two different models are exactly solved in the second one a nonequilibrium bimodal distribution induced by the noise is observed for a critical value of its correlation time critical slowing down does not appear in this point but in another one

improving the accuracy of the discrete gradient method in the one dimensional case we present two numerical schemes of high accuracy for one dimensional dynamical systems they are modifications of the discrete gradient method and keep its advantages including stability and conservation of the energy integral however their accuracy is higher by several orders of magnitude

one dimensional hyperbolic transport positivity and admissible boundary conditions derived from the wave formulation abstract we consider the one dimensional cattaneo equation for transport of scalar fields such as solute concentration and temperature in mass and heat transport problems respectively although the cattaneo equation admits a stochastic interpretation at least in the one dimensional case negative concentration values can occur in boundary value problems on a finite interval this phenomenon stems from the probabilistic nature of this model the stochastic interpretation provides constraints on the admissible boundary conditions as can be deduced from the wave formulation here presented moreover as here shown energetic inequalities and the dissipative nature of the equation provide an alternative way to derive the same constraints on the boundary conditions derived by enforcing positivity the analysis reported is also extended to transport problems in the presence of a biasing velocity field several general conclusions are drawn from this analysis that could be extended to the higher dimensional case

behavior of entropy in hyperbolic heat conduction it is shown that the classical entropy of an isolated system does not increase monotonically when the approach to equilibrium is described by hyperbolic equations for the temperature evolution in contrast the generalized entropy of extended irreversible thermodynamics does increase monotonically and is therefore better suited than the classical theory for the description of hyperbolic heat transport

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

geometric exponential integrators abstract in this paper we consider exponential integrators for semilinear poisson systems two types of exponential integrators are constructed one preserves the poisson structure and the other preserves energy numerical experiments for semilinear possion systems obtained by semi discretizing hamiltonian pdes are presented these geometric exponential integrators exhibit better long time stability properties as compared to non geometric integrators and are computationally more efficient than traditional symplectic integrators and energy preserving methods based on the discrete gradient method

construction of runge kutta type methods for solving ordinary differential equations in this paper we study runge kutta methods with continuous stage introduced by butcher in 1987 by setting the coefficients of this family of methods and choosing the appropriate numerical integration we derive new classes of runge kutta methods furthermore we extend the w transformation technique by permitting w to be a non square matrix this allows us to construct more high order implicit runge kutta methods with some geometric properties specially we provide runge kutta methods with continuous stage which are conjugate symplectic symmetric or energy preserving for solving hamiltonian systems we also present some numerical experiments to verify our results

nonstationary markovian replication process causing diverse diffusions we introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions a nonstationary markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker s displacement and the generalized telegrapher equation with time varying coefficients and we find that diffusivity can be determined by temporal changes of replication of an immediate step by controlling the replications we realize diverse diffusions such as alternating diffusion superdiffusion subdiffusion and marginal diffusion which originate from oscillating increasing decreasing and slowly increasing or decreasing replications with time respectively

on random flights with non uniformly distributed directions this paper deals with a new class of random flights in d d 2 characterized by non uniform probability distributions on the multidimensional sphere these random motions differ from similar models appeared in literature where the directions are taken according to the uniform law the family of angular probability distributions introduced in this paper depends on a parameter 0 which gives the anisotropy of the motion furthermore we assume that the number of changes of direction performed by the random flight is fixed the time lengths between two consecutive changes of orientation have joint probability distribution given by a dirichlet density function

completely conservative covariant numerical methodology for a distance dependent potential o newtonian dynamics is characterized by conservation of energy linear momentum and angular momentum in addition the basic dynamical equations are covariant that is invariant under fundamental coordinate transformations in this paper it is shown that related n body problems can be solved numerically in such a fashion that independently of the time step the identical conservation laws continue to be valid for the approximating difference equations in addition the numerical method is covariant

telegrapher s equations with variable propagation speeds all derivations of the one dimensional telegrapher s equation based on the persistent random walk model assume a constant speed of signal propagation we generalize here the model to allow for a variable propagation speed and study several limiting cases in detail we also show the connections of this model with anomalous diffusion behavior and with inertial dichotomous processes

generating functions for dynamical systems with symmetries integrals and differential invariants abstract we give a survey and some new examples of generating functions for systems with symplectic structure systems with a first integral systems that preserve volume and systems with symmetries and or time reversing symmetries both odes and maps are treated and we discuss how generating functions may be used in the structure preserving numerical integration of odes with the above properties

classical field theory and stochastic properties of hyperbolic equations of dissipative processes abstract the set of damped hyperbolic transport equations is one of the wide class of equations for the description of dissipative physical processes deeper understanding into the structure of these physical phenomena can be obtained with the help of the hamiltonian formalism in the present paper we show that the hamilton lagrange formalism can be constructed for these kinds of transport equations we obtain the hamiltonian the canonically conjugate quantities and the poisson bracket expressions for them with this formalism we analyze the statistical properties of path fluctuations in the new conjugated thermodynamic variable space we show that for short times the stochastic behavior under this new scope obeys the chapman kolmogorov relationship

discrete gradient approach to stochastic differential equations with a conserved quantity general autonomous stochastic differential equations sdes driven by one dimensional brownian motion in the stratonovich sense with a conserved quantity i y are considered relying on this conserved quantity an equivalent skew gradient sg form of original sdes is constructed with the aim of constructing conserved numerical methods direct discrete gradient approaches which approximate directly the sg system lead to two conserved methods in indirect discrete gradient approaches we first split the sg system into subsystems which preserve the original conserved quantity then apply direct discrete gradient approaches to subsystems to produce two composition schemes the mean square convergence of order 1 for these four methods depends on the assumptions that the coefficients of sdes are lipschitz continuous with bounded second moments along the solution of sdes are twice continuous differentiable and have derivatives that are uniformly bounded three numerical experiments are presented to verify our theoretical analysis and show the advantages of these methods

long time behaviour of discretizations of the simple pendulum equation we compare several discretizations of the simple pendulum equation in a series of numerical experiments the stress is put on the long time behaviour the chosen numerical schemes are either symplectic maps or integrable energy preserving maps or both therefore they preserve qualitative features of solutions such as periodicity we describe characteristic periodic time dependences of numerical estimates of the period and the amplitude and explain them as systematic numerical by effects produced by any method finally we propose a new numerical scheme which is a modification of the discrete gradient method this modified discrete gradient method preserves almost exactly the period of small oscillations for any time step