order conditions for canonical runge kutta schemes when numerically integrating hamiltonian systems of differential equations it is often advantageous to use canonical methods i e methods that preserve the symplectic structure of the phase space thus reproducing an important feature of the hamiltonian flow an s stage runge kutta rk method without redundant stages is canonical if and only if with a standard notation b i a ij b j a ji b i b j 0 1 leqq i j leqq s it is shown that for canonical rk methods there are many redundancies in the standard order conditions for a canonical method to have order p it is sufficient that the b i s a ij s satisfy a system of algebraicequations that has roughly speaking an equation per nonrooted tree of order leqq p furthermore a new methodology is presented for the investigation of the order of canonical integration methods not necessarily rk methods when applied to hamiltonian systems in the new approach consistency is studied by taylor expanding a suitable scalar function in ter

discretising the herman kluk propagator the herman kluk propagator is a well known semi classical approximation of the unitary evolution operator in quantum molecular dynamics in this paper we formulate the herman kluk propagator as a phase space integral and discretise it by monte carlo and quasi monte carlo quadrature then we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with fourier integral operator calculus numerical experiments for two and six dimensional model systems support our theoretical results

unconventional schemes for a class of ordinary differential equations with applications to the korteweg de vries equation an unconventional numerical method for solving a restrictive and yet often encountered class of ordinary differential equations is proposed the method has a crucial what we callreflexive property and requires solving one linear system per time step but is second order accurate a systematical and easily implementable scheme is proposed to enhance the computational efficiency of such methods whenever needed applications are reported on how the idea can be applied to solve the korteweg de vries equation discretized in space

high order time splitting hermite and fourier spectral methods in this paper we are concerned with the numerical solution of the time dependent gross pitaevskii equation gpe involving a quasi harmonic potential primarily we consider discretisations that are based on spectral methods in space and higher order exponential operator splitting methods in time the resulting methods are favourable in view of accuracy and efficiency moreover geometric properties of the equation such as particle number and energy conservation are well captured regarding the spatial discretisation of the gpe we consider two approaches in the unbounded domain we employ a spectral decomposition of the solution into hermite basis functions on the other hand restricting the equation to a sufficiently large bounded domain fourier techniques are applicable for the time integration of the gpe we study various exponential operator splitting methods of convergence orders two four and six our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting fourier and hermite pseudospectral methods for the time evolution of the gpe furthermore we illustrate the effectiveness of higher order time splitting methods compared to standard integrators in a long term integration

composition constants for raising the orders of unconventional schemes for ordinary differential equations many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general purpose software designed to solve numerically a wide range of differential equations if those properties are to be exploited fully for the sake of better numerical stability accuracy and or speed the differential equations may have to be solved by unconventional methods this short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods called reflexive

sound penetration through an arbitrarily shaped aperture in a soft screen analytical approach van bladel has reduced the problem of low frequency scattering through an aperture in a soft screen to a sequence of static integral equations analytical solutions are known at the moment for a circular and an elliptic aperture only a new analytical method is proposed here which is valid for the apertures of arbitrary shape the method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier specific approximate formulae are derived for evaluating the average value of the zeroth order term in the low frequency expansion for an aperture in the shape of a polygon a triangle a rectangle a rhombus a circular sector and a circular segment all the formulae are checked against the numerical solutions known in the literature and a good agreement is confirmed

sound transmission through an arbitrarily shaped aperture in a rigid screen analytical approach van bladel has reduced the problem of low frequency scattering through an aperture in a rigid screen to a sequence of static integral equations analytical solutions are known at the moment for a circular and an elliptic aperture only a new analytical method is proposed here which is valid for apertures of arbitrary shape the method is based on an integral representation for the reciprocal distance between two points which was previously obtained by the author specific approximate formulae are derived for evaluating the average value of the first term in the low frequency expansion for an aperture in the shape of a polygon a triangle a rectangle a rhombus a circular sector and a circular segment all the formulae are checked against the numerical solutions known in the literature and a good accuracy is confirmed the second and the third terms of the low frequency expansion can be derived in a similar manner and the related formulae are to be publlshed later

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

on the numerical integration of ordinary differential equations by processed methods we provide a theoretical analysis of the processing technique for the numerical integration of odes we get the effective order conditions for processed methods in a general setting so that the results obtained can be applied to different types of numerical integrators we also propose a procedure to approximate the postprocessor such that its evaluation is virtually cost free the analysis is illustrated for a particular class of composition methods

hybrid solvers for composition and splitting methods composition and splitting are useful techniques for constructing special purpose integration methods for numerically solving many types of differential equations in this article we will review these methods and summarise the essential ingredients of an implementation that has recently been added to a framework for solving differential equations in mathematica

implementation of high order spline interpolations for tracking test particles in discretized fields a systematic approach for constructing high order spline interpolation methods is proposed for fields known on regular rectangular grids these interpolation methods are tested in tracking trajectories of particles submitted to a force that derives from a potential known on a grid the interplay between the time advancement scheme and the spatial interpolation is studied in detail and it is shown how the order of the trajectory solver is directly affected by the order of the spline interpolation it is also shown how an interpolation method that preserves topological properties of physical fields can be better exploited with these higher order spline approximations

computer evaluation of singular integrals and their applications an algorithm and a standard subroutine are developed for the evaluation of singular integrals over arbitrary two dimensional domains the integrand is a product of a green s function with another function having a weak singularity at the boundary of the domain formulae are derived for an accurate estimation of the integral in the neighbourhood of the singularities the integral over the rest of the domain is evaluated by a library subroutine the software developed is applied to a study of some contact problems of the theory of elasticity for non classical domains

hydrodynamic friction and the capacitance of arbitrarily shaped objects the translational friction coefficient and the capacitance of a variety of objects are calculated with a probabilistic method involving hitting the probed objects with random walks launched from an enclosing spherical surface this method is applied to exactly solvable examples to test the program accuracy and to physically important and analytically intractable examples cube chain of spheres at the vertices of self avoiding and random walks etc large fluctuations in the friction of polymer chains with a random coil structure are found to give large deviations from the mean field kirkwood riseman theory and hydrodynamic fluctuation effects are found to diminish with the chain swelling accompanying excluded volume interaction capacity applications are reviewed and our probabilistic estimates of polymer friction are compared with previous calculations using alternative methods transients to the capacity and related properties are expressed in terms of fluctuations in the wiener sausage volume volume swept out by a brownian particle where a repeated visit to a spatial region does not contribute to the volume increase in time

a family of anadromic numerical methods for matrix riccati differential equations matrix riccati differential equations mrdes x a21 xa11 a22x xa12x x 0 x0 where aij aij t appear frequently throughout applied mathematics science and engineering naturally the existing conventional runge kutta methods and linear multi step methods can be adapted to solve mrdes numerically indeed they have been adapted there are a few unconventional numerical methods too but they are suited more for time invariant mrdes than time varying ones for stiff mrdes existing implicit methods which are preferred to explicit ones require solving nonlinear systems of equations of possibly much higher dimensions than the original problem itself of for example implicit runge kutta methods and thus they can pose implementation difficulties and also be expensive in the past the property of an mrde which has been most preserved is the symmetry property for a symmetric mrde many other crucial properties have been discarded besides the symmetry property our proposed methods also preserve two other important properties bilinear rational dependence on the initial value and a generalized inverse relation between an mrde and its complementary mrde by preserving the generalized inverse relation our methods are accurately able to integrate an mrde whose solution has singularities by preserving the property of bilinear dependence on the initial value our methods also conserve the rank of change to the initial value and a solution s monotonicity property our methods are anadromic 1meaning if an mrde is integrated by one of our methods from t to and then integrated backward from t to using the same method the value at t is recovered in the absence of rounding errors this implies that our methods are necessarily of even order of convergence for time invariant mrdes methods of any even order of convergence are established while for time varying mrdes methods of order as high as 10 are established but only methods of order up to 6 are stated in detail our methods are semi implicit in the sense that there are no nonlinear systems of matrix equations to solve only linear ones unlike any pre existing implicit method given the availability of high quality codes for linear matrix equations our methods can easily be implemented and embedded into any application software package that needs a robust mrde solver numerical examples are presented to support our claims 1we coined anadromic from the greek roots back up and o o act of running of anadromous which describes fish that must return from the ocean to spawn in the same streams whence they hatched after we had tried to describe our numerical methods with words such as reflexive 27 28 symmetric which is still used today in the literature e g 22 and reversible which turned out to be overused received by the editor received august 25 2009 and in revised form october 30 2010 2010 mathematics subject classification primary 65l05

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

on the connection between the malliavin covariance matrix and hormander s condition abstract a celebrated theorem of hormander gives a sufficient condition for a second order differential operator to be hypoelliptic for operators with analytic coefficients this condition turns out to be also necessary but this is not true for general smooth coefficients on the other hand malliavin conceived a probabilistic approach to the same problem known as malliavin calculus in which a key role is played by the malliavin covariance matrix the aim of our paper is to give several characterizations of the malliavin covariance matrix which are equivalent to hormander s condition and consequently imply the hypoellipticity in this way the distance between hormander s condition and the hypoellipticity property is clearly pointed out in probabilistic terms

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

composition methods for differential equations with processing we construct numerical integrators for differential equations up to order 12 obtained by composition of basic integrators the following cases are considered i composition for a system separable in two solvable parts ii composition using as basic methods a first order integrator and its adjoint iii composition using second order symmetric methods and iv composition using fourth order symmetric methods each scheme is implemented with a processor or corrector to improve their efficiency and this can be done virtually cost free

a structure preserving flow for computing hamiltonian matrix exponential this article focuses on computing hamiltonian matrix exponential given a hamiltonian matrix mathcal h it is well known that the matrix exponential e mathcal h is a symplectic matrix and its eigenvalues form reciprocal lambda 1 bar lambda it is important to take care of the symplectic structure for computing e mathcal h based on the structure preserving flow proposed by kuo et al siam j matrix anal appl 37 976 1001 2016 we develop a numerical method for computing the symplectic matrix pair mathcal m mathcal l which represents e mathcal h