exponential power series expansion for the propagator of general diffusion processes abstract the propagator of a general diffusion process is determined by expanding its exponent in a power series of a time increment t the expansion coefficients can be analytically evaluated from recursive relations we are thus able to construct a covariant short time approximation for the propagator valid to any desired precision in t attention is given both to the mathematics and its physical interpretation this allows us to shed further light on the results already known in the literature on quantum mechanics and theory of continuous markov processes

quantum monte carlo methods recent developments abstract recent developments of the quantum monte carlo methods are reviewed briefly with emphasis on basic ideas namely the transformation of a d dimensional quantum system to the corresponding d 1 dimensional classical system some recent applications to quantum spin systems are summarized in particular numerical evidence of the haldane gap is given the quantum transfer matrix method is also presented for it is also based on the st transformation the quantum monte carlo cam is explained briefly fractal path integrals are also presented namely a general theory of decomposition of exponential operators and symplectic integrators are explained the negative sign problem is also discussed

a split step numerical method for the time dependent dirac equation in 3 d axisymmetric geometry abstract a numerical method is developed to solve the time dependent dirac equation in cylindrical coordinates for 3 d axisymmetric systems the time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration while the wave function is discretized spatially on a uniform grid the longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses poisson s integral solution which allows to implement the radial symmetry of the wave function the latter is evaluated on a time staggered mesh by using hermite polynomial interpolation and by performing the integration analytically the cylindrical coordinate singularity problem at r 0 is circumvented by this method as the integral is well defined at the origin the resulting scheme is reminiscent of non standard finite differences in the last step of the splitting the remaining equation has a solution in terms of a time ordered exponential which is approximated to a higher order than the time evolution scheme we study the time evolution of gaussian wave packets and we evaluate the eigenstates of hydrogen like systems by using a spectral method we compare the numerical results to analytical solutions to validate the method in addition we present three dimensional simulations of relativistic laser matter interactions using the dirac equation

decomposition formulas of exponential operators and lie exponentials with some applications to quantum mechanics and statistical physics decomposition formulas of general exponential operators in a banach algebra and in a lie algebra are presented that yield a basis of monte carlo simulation of quantum systems they are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena a global approximation method of transient phenomena is also formulated on the basis of decomposition formulas it is applied to the laser model as a simple example

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

approximating the exponential from a lie algebra to a lie group consider a differential equation y a t y y y 0 y0 with y 0 g and a r g g where g is a lie algebra of the matricial lie group g every b g can be mapped to g by the matrix exponential map exp tb with t r most numerical methods for solving ordinary differential equations odes on lie groups are based on the idea of representing the approximation y n of the exact solution y t n t n r by means of exact exponentials of suitable elements of the lie algebra applied to the initial value y 0 this ensures that y n g when the exponential is difficult to compute exactly as is the case when the dimension is large an approximation of exp tb plays an important role in the numerical solution of odes on lie groups in some cases rational or polynomial approximants are unsuitable and we consider alternative techniques whereby exp tb is approximated by a product of simpler exponentials in this paper we present some ideas based on the use of the strang splitting for the approximation of matrix exponentials several cases of g and g are considered in tandem with general theory order conditions are discussed and a number of numerical experiments conclude the paper

analysis of robust multigrid methods for steady viscous low mach number flows a theoretical and numerical analysis of different implicit methods in multigrid form is carried out in order to study their behaviour at various flow conditions the fourier analysis for scalar convection diffusion equations is extended to the coupled set of laminar navier stokes equations for first and second order discretized navier stokes equations the analysis of point line and block methods is in good agreement with the numerical results the influence on the convergence rate of different second order implementations in the multigrid method is also considered the combination of multigrid line methods and mixed discretization enables to tackle stiff problems in a cheap and robust way the comparison is further extended to reynolds averaged navier stokes equations with the low reynoldsk equations describing the turbulence when line and block methods are used it is advantageous not to incorporate the turbulence equations into the multigrid cycle

multi staging of jacobi relaxation to improve smoothing properties of multigrid methods for steady euler equations abstract multi stage versions of jacobi relaxation are studied for use in multigrid methods for steady euler equations it is shown that these multi stage versions allow much more general and much more efficient multigrid methods than possible with classic relaxation methods

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

variational quantum simulation of general processes variational quantum algorithms have been proposed to solve static and dynamic problems of closed many body quantum systems here we investigate variational quantum simulation of three general types of tasks generalized time evolution with a non hermitian hamiltonian linear algebra problems and open quantum system dynamics the algorithm for generalized time evolution provides a unified framework for variational quantum simulation in particular we show its application in solving linear systems of equations and matrix vector multiplications by converting these algebraic problems into generalized time evolution meanwhile assuming a tensor product structure of the matrices we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution finally we introduce variational quantum simulation for open system dynamics we variationally implement the stochastic schrodinger equation which consists of dissipative evolution and stochastic jump processes we numerically test the algorithm with a 6 qubit 2d transverse field ising model under dissipation

product formulas for exponentials of commutators we provide a recursive method for systematically constructing product formula approximations to exponentials of commutators giving approximations that are accurate to arbitrarily high order using these formulas we show how to approximate unitary exponentials of possibly nested commutators using exponentials of the elementary operators and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance by presenting an algorithm for quantum search using evolution according to a commutator we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal finally we discuss applications of our product formulas to quantum control and to implementing anticommutators providing new methods for simulating many body interaction hamiltonians

existence of short time approximations of any polynomial order for the computation of density matrices by path integral methods in this paper i provide significant mathematical evidence in support of the existence of direct short time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials while for theorem 2 which is experimental i only provide a physicist s proof i believe the present development is mathematically sound as a verification i explicitly construct two short time approximations to the density matrix having convergence orders 3 and 4 respectively furthermore in appendix b i derive the convergence constant for the trapezoidal trotter path integral technique the convergence orders and constants are then verified by numerical simulations while the two short time approximations constructed are of sure interest to physicists and chemists involved in monte carlo path integral simulations the present paper is also aimed at the mathematical community who might find the results interesting and worth exploring i conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real time feynman path integral simulations

study of correction terms for higher order decompositions of exponential operators many kinds of higher order decompositions are applied to an exactly solvable model and the trotter number dependence of the correction terms is evaluated systematically practically useful decompositions are explored

multigrid acceleration of a flux difference splitting method for steady euler equations abstract a flux difference splitting based on the polynomial character of the flux vectors is introduced for steady euler equations this splitting is applied to finite volumes centered around the vertices of the computational grid a discrete set of equations is obtained which is both conservative and positive the flux difference splitting is done in an algebraically exact way as a consequence shocks are represented as sharp discontinuities without wiggles due to the positivity the set of equations can be solved by collective relaxation methods a full multigrid method based on symmetric successive relaxation full weighting bilinear interpolation and w cycle is presented typical full multigrid efficiency is achieved for the gamm transonic bump test case since after the starting cycle and one multigrid cycle the solution cannot be distinguished anymore from the fully converged solution

cellular automaton approach to electromagnetic wave propagation in dispersive media extensions of bialynicki birula s cellular automaton are proposed for studies of the onedimensional propagation of electromagnetic fields in drude metals as well as in both transparent dispersive and lossy dielectrics these extensions are obtained by representing the dielectrics with appropriate matter fields such as polarization together with associated velocity fields to obtain the different schemes for the integration of the resulting systems of linear partial differential equations split operator ideas are employed possible further extensions to two dimensional propagation and for the study of left handed materials are discussed the stability properties of the cellular automaton treated as a difference scheme are analysed

refined formulation of quantum analysis q derivative and exponential splitting quantum analysis is reformulated to clarify its essence namely the invariance of quantum derivative for any choice of definitions of the differential df a satisfying the leibniz rule this formulation with use of the inner derivation a is convenient to study quantum corrections in contrast to the feynman operator calculus the present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order general recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae multiple integral representations of q derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators a simple explanation of the connection between quantum derivatives and q derivative is also given

fractal path integrals with applications to quantum many body systems abstract a general formulation of the fractal decomposition of exponential operators is reviewed briefly and its mathematical structure is discussed together with some applications to quantum many body systems more explicitly the operator e x a b is decomposed into a product of the form ex a b f m x o x m 1 with f m x e t 1 xa e f 2 xb e f 3 xa e t 4 xb e t m xb the parameters t j show a fractal structure as shown in the text

compact exponential product formulas and operator functional derivative a new scheme for deriving compact expressions of the logarithm of the exponential product is proposed and it is applied to several exponential product formulas a generalization of the dynkin specht wever dsw theorem on free lie elements is given and it is used to study the relation between the traditional method based on the dsw theorem and the present new scheme the concept of the operator functional derivative is also proposed and it is applied to ordered exponentials such as time evolution operators for time dependent hamiltonians

stability of the trotter suzuki decomposition the trotter suzuki decomposition tsd is an important tool for the simulation and control of physical systems we provide evidence for the stability of the tsd we model the error in the decomposition and determine sufficiency conditions that guarantee the stability of this decomposition under this model we relate these sufficiency conditions to precision limitations of computing and control in both classical and quantum cases furthermore we show that bounded error tsd can be achieved by a suitable choice of machine precision

an efficient algorithm for translationally invariant finite size periodic lattice systems in one spatial dimension we develop an efficient algorithm to find a matrix product state representation of the ground state wavefunctions for translationally invariant finite size periodic lattice systems in one spatial dimension this is based on the observation that the efficient computation of the ground state energy per site only needs to retain a certain number of the largest eigenvalues of the transfer matrix for a matrix product state without any sacrifice of accuracy the computational cost is independent of the system s size and scales as 3 with being the truncation dimension the algorithm is tested for the critical quantum ising model in a transverse field on a finite size lattice with the size as large as 4800 for the truncation dimension 200

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

the gaussian wave packet transform efficient computation of the semi classical limit of the schrodinger equation part 1 formulation and the one dimensional case in this paper an efficient and accurate method for simulating the propagation of a localized solution of the schrodinger equation with a smooth potential near the semiclassical limit is presented we are interested computing arbitrarily accurate solutions when the non dimensional planck s constant e is small but not negligible the method is based on a time dependent transformation of the independent variables closely related to gaussian wave packets a rescaled wave function w satisfies a new schrodinger equation with a time dependent potential which is a perturbation of the harmonic oscillator the perturbation being o e so that all stiffness in space and time is greatly reduced in fact for integration in a fixed time interval the number of modes required to fully resolve the problem decreases when e is decreased the original wave function may be reconstructed by fourier interpolation although expectation values of the observables can be computed directly from the function w itself if the initial condition is a gaussian wave packet very few modes are necessary to fully resolve the w variable so for short time very accurate solutions can be obtained at low computational cost initial conditions other than gaussians wave packets can also be used in this paper the gaussian wave packet transform is carefully outlined and applied to the schrodinger equation in one dimension detailed numerical tests show the efficiency and accuracy of the approach in the sequel of this paper this approach has been extended to the multidimensional case

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

hamiltonian formulation of energy conservative variational equations by wavelet expansion abstract hamiltonian formulation of various energy conservative evolution equations is given by means of wavelet expansion of solutions on the whole real axis r the kdv equation wave equations and schrodinger equations are treated in a unified similar manner a matrix representation of operators with respect to a nice wavelet base plays an important role in the formulation since the procedure is very concrete our results can be used to efficiently compute numerical solutions of partial differential equations described in the text in fact we may also use symplectic schemes to solve derived hamiltonian systems

quantum analysis non commutative differential and integral calculi a new scheme of quantum analysis namely a non commutative calculus of operator derivatives and integrals is introduced this treats differentiation of an operator valued function with respect to the relevant operator in a banach space in this new scheme operator derivatives are expressed in terms of the relevant operator and its inner derivation explicitly derivatives of hyperoperators are also defined some possible applications of the present calculus to quantum statistical physics are briefly discussed

convergence of general decompositions of exponential operators a general theorem is proved concerning the convergence of decompositions of exponential operators in a banach space or normed space as a corollary the convergence of fractal decompositions is proved the convergence of generalized trotter like formulas is also shown to result from the general theorem

on the configurational temperature nose hoover thermostat abstract in this paper we reformulate the configurational temperature nose hoover thermostat of braga and travis 2005 by means of a quasi hamiltonian theory in phase space sergi and ferrario 2001 the quasi hamiltonian structure is exploited to introduce a hybrid configurational kinetic temperature nose hoover chain thermostat that can achieve a uniform sampling of phase space also for stiff harmonic systems as illustrated by simulating the dynamics of one dimensional harmonic and quartic oscillators an integration algorithm based on the symmetric trotter decomposition of the propagator is presented and tested against implicit geometric algorithms with a structure similar to the velocity and position verlet in order to obtain an explicit form for the symmetric trotter propagator algorithm in the case of non harmonic and non linear interaction potentials a position dependent harmonically approximated propagator is introduced such a propagator approximates the dynamics of the configurational degrees of freedom as if they were locally moving in a harmonic potential the resulting approximated locally harmonic dynamics is tested with good results in the case of a one dimensional quartic oscillator the integration is stable and locally time reversible instead the implicit geometric integrator is stable and time reversible globally when convergence is achieved we also verify the stability of the approximated explicit integrator for a three dimensional n particle system interacting through a soft weeks chandler andersen potential

high order non unitary split step decomposition of unitary operators we propose a high order numerical decomposition of exponentials of hermitian operators in terms of a product of exponentials of simple terms following an idea which has been pioneered by m suzuki implementing it for complex coefficients we outline a convenient fourth order formula which can be written compactly for an arbitrary number of non commuting terms in the hamiltonian and which is superior to the optimal formula with real coefficients both in complexity and accuracy we show asymptotic stability of our method for a sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models

structure of positive decompositions of exponential operators the solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data accurate solutions can be systematically derived by decomposing the exponential in a product form for time reversible equations such as the hamilton or the schr odinger equation it is immaterial whether or not the decomposition coefficients are positive in fact most symplectic algorithms for solving classical dynamics contain some negative coefficients for time irreversible systems such as the fokker planck equation or the quantum statistical propagator only positive coefficient decompositions which respect the time irreversibility of the diffusion kernel can yield practical algorithms these positive time steps only forward decompositions are a highly effective class of factorization algorithms this work presents a framework for understanding the structure of these algorithms by a suitable representation of the factorization coefficients we show that specific error terms and order conditions can be solved analytically using this framework we can go beyond the sheng suzuki theorem and derive a lower bound for the error coefficient e vtv by generalizing the framework perturbatively we can further prove that it is not possible to have a sixth order forward algorithm by including only the commutator vtv ensuremath equiv mathbf v t v mathbf the pattern of these higher order forward algorithms is that in going from the 2n mathrm th to the 2n 2 mathrm th order one must include a different commutator v t 2n ensuremath 1 v in the decomposition process

multigrid formulation of polynomial flux difference splitting for steady euler equations a flux difference splitting method based on the polynomial character of the flux vectors is applied to steady euler equations the discretization is done with a vertex centered finite volume method in first order form a discrete set of equations is obtained which is both conservative and positive the flux difference splitting is done in an algebraically exact way so that shocks are represented without wiggles due to the positivity the set of equations can be solved by collective relaxation methods a full multigrid method based on symmetric successive relaxation full weighting bilinear interpolation and w cycle is presented in firstorder form typical full multigrid efficiency is achieved this is demonstrated on the gamm transonic bump test case the second order formulation is based on the roe chakravarthy minmod limiter the discrete system is solved using a multigrid defect correction formulation the second order formulation is demonstrated on harten s shock reflection problem

forward symplectic integrators and the long time phase error in periodic motions we show that when time reversible symplectic algorithms are used to solve periodic motions the energy error after one period is generally two orders higher than that of the algorithm by use of correctable algorithms we show that the phase error can also be eliminated two orders higher than that of the integrator the use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth order accuracy in the periodic energy we study the one dimensional harmonic oscillator and the two dimensional kepler problem in great detail and compare the effectiveness of some recent fourth order algorithms

discontinuous galerkin hp adaptive methods for multiscale chemical reactors quiescent reactors abstract we present a class of chemical reactor systems modeled numerically using a fractional multistep method between the reacting and diffusing modes of the system subsequently allowing one to utilize algebraic techniques for the resulting reactive subsystems a mixed form discontinuous galerkin method is presented with implicit and explicit imex timestepping strategies coupled to dioristic entropy schemes for h p adaptivity of the solution where the h and p are adapted based on an l 1 stability result finally we provide some numerical studies on the convergence behavior adaptation and asymptotics of the system applied to a pair of equilibrium problems as well as to general three dimensional nonlinear lotka volterra chemical systems

a liouville operator derived measure preserving integrator for molecular dynamics simulations in the isothermal isobaric ensemble the constant pressure constant temperature npt molecular dynamics approach is re examined from the viewpoint of deriving a new measure preserving reversible geometric integrator for the equations of motion the underlying concepts of non hamiltonian phase space analysis measure preserving integrators and the symplectic property for hamiltonian systems are briefly reviewed in addition current measure preserving schemes for the constant volume constant temperature ensemble are also reviewed a new geometric integrator for the npt method is presented is shown to preserve the correct phase space volume element and is demonstrated to perform well in realistic examples finally a multiple time step version of the integrator is presented for treating systems with motion on several time scales

complete characterization of fourth order symplectic integrators with extended linear coefficients the structure of symplectic integrators up to fourth order can be completely and analytically understood when the factorization split coefficients are related linearly but with a uniform nonlinear proportional factor the analytic form of these extended linear symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and nonforward fourth order algorithms with an arbitrary number of operators most fourth order forward integrators can now be derived analytically from this extended linear formulation without the use of symbolic algebra

a flux difference splitting method for steady euler equations abstract the polynomial flux difference splitting method is applied to steady euler equations a discrete set of equations which is both conservative and positive is obtained the set of equations is solved by the relaxation technique

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

nearly optimal lattice simulation by product formulas we consider simulating an n qubit hamiltonian with nearest neighbor interactions evolving for time t on a quantum computer we show that this simulation has gate complexity nt 1 o 1 using product formulas a straightforward approach that has been demonstrated by several experimental groups while it is reasonable to expect this complexity in particular this was claimed without rigorous justification by jordan lee and preskill we are not aware of a straightforward proof our approach is based on an analysis of the local error structure of product formulas as introduced by descombes and thalhammer and significantly simplified here we prove error bounds for canonical product formulas which include well known constructions such as the lie trotter suzuki formulas we also develop a local error representation for time dependent hamiltonian simulation and we discuss generalizations to periodic boundary conditions constant range interactions and higher dimensions combined with a previous lower bound our result implies that product formulas can simulate lattice hamiltonians with nearly optimal gate complexity

hyperbolicity and optimal coordinates for the three dimensional supersonic euler equations the two dimensional 2 d steady euler equations for supersonic flow in cartesian coordinates x y is hyperbolic except on the singular line u a where a is the speed of sound and u is the x component of velocity however the cauchy problem for x x circ with initial data prescribed at x x circ is well posed only when u a for the 2 d case the optimal coordinates have been found w h hui and d l chu comput fluid dynamics 4 1996 pp 403 426 to be the orthogonal system consisting of streamlines and their orthogonal lines this gives the most robust and accurate computation for supersonic flow it is commonly thought e g c y loh and m s liou j comput phys 113 1994 pp 224 248 that the system of three dimensional 3 d steady euler equations for supersonic flow is hyperbolic the present paper shows that it is hyperbolic only when the velocity in the marching direction is supersonic in which case it also simultaneously guarantees the well posedness of the cau

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

a split step approach for the 3 d maxwell s equations split step procedures have previously been used successfully in a number of situations e g for hamiltonian systems such as certain nonlinear wave equations in this study we note that one particular way to write the 3 d maxwell s equations separates these into two parts requiring only the solution of six uncoupled 1 d wave equations the approach allows arbitrary orders of accuracy in both time and space and features in many cases unconditional stability

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

optimized high order splitting methods for some classes of parabolic equations we are concerned with the numerical solution obtained by splitting methodsof certain parabolic partial differential equations splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients it has been demonstrated that this se cond order barrier can be overcome by using splitting methods with complex valued coefficients with positive real parts in this way method s of orders 3 to 14 by using the suzuki yoshida triple and quadruple jump composition procedure have been explicitly built here we reconsider this technique an d show that it is inherently bounded to order 14 and clearly sub optimal with respect to error constants as an alternative we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time even when low accuracies are desired we also show that in the general case 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8

approximations to di and tri logarithms we propose hypergeometric constructions of simultaneous approximations to polylogarithms these approximations suit for computing the values of polylogarithms and satisfy 4 term apery like polynomial recursions

time discretization of functional integrals numerical evaluation of functional integrals usually involves a finite l slice discretization of the imaginary time axis in the auxiliary field method the l slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite l as l 1 l l if the density matrix 1 in the static approximation is known we investigate the convergence of the partition function zl tr l the internal energy and the density of states gl e the inverse laplace transform of zl as l for the simple harmonic oscillator gl e is a normalized truncated fourier series for the exact density of states when the auxiliary field approach is applied to spin systems approximants to the density of states and heat capacity can be negative approximants to the density matrix for a spin dimer are found in closed form for all l by appending a self interaction to the divergent gaussian integral and analytically continuing to zero self interaction because of this continuation the coefficient of the singlet projector in the approximate density matrix can be negative for a spin dimer zl is an even function of the coupling constant for l 3 ferromagnetic and antiferromagnetic coupling can be distinguished only for l 3 where a berry phase appears in the functional integral at any non zero temperature the exact partition function is recovered as l

convergence of unitary and complex decompositions of exponential operators the convergence of the fractal unitary and complex decompositions of exponential operators is proved for hermitian operators in a banach space

higher order decompositions of ordered operator exponentials we present a decomposition scheme based on lie trotter suzuki product formulae to approximate an ordered operator exponential with a product of ordinary operator exponentials we show using a counterexample that lie trotter suzuki approximations may be of a lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order to address this problem we present a set of criteria that is sufficient for the validity of these approximations prove convergence and provide upper bounds on the approximation error this work may shed light on why related product formulae fail to be as accurate as expected when applied to coulomb potentials

goldberg s theorem and the baker campbell hausdorff formula abstract goldberg s theorem duke math j 23 1956 13 is applied to expressing the logarithm of exponential product in terms of free lie elements using hyperoperators namely inner derivations a symmetric property of the expansion coefficients in the formal power series is also discussed

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

rational approximations to the dilogarithm the irrationality proof of the values of the dilogarithmic function l 2 z at rational points z 1 k for every integer k 5 7 is given to show this we develop the method of pade type approximations using legendre type polynomials which also derives good irrationality measures of l 2 1 k moreover the linear independence over q of the numbers 1 log 1 1 k and l 2 1 k is also obtained for each integer k 5 7

a lattice gas automaton capable of modeling three dimensional electromagnetic fields a lattice gas automaton lga capable of modeling maxwell s equations in three dimensions is described the automaton is a three dimensional interconnection of two dimensional lga cells with appropriate operations at the junctions between cells to include the properties of polarization a homogeneous mathematical description of the heterogeneous three dimensional automaton is provided in terms of the underlying binary variables the dynamics of the automaton conserve the scalar components of the electric and magnetic fields the implementation of the automaton on the cam 8 cellular automata machine is described the lga has been validated through calculation of resonant frequencies within various cavities the numerical results indicate the success of the automaton in analyzing three dimensional em field problems we have not proven analytically that this model reproduces maxwell s equations in the macroscopic limit as this is a topic of future study

diophantine properties of numbers related to catalan s constant 1 as a linear form in 1 and even beta valuesonly the following lemma 2 is almost the same as the correspondingone in br and we only include a sketch of the proof we would like toemphasise the fact that it is essentially the well poisedness of the series thatenables us to produce this parity dependant behaviour when

a new range reduction algorithm range reduction is a key point for getting accurate elementary function routines we introduce a new algorithm that is fast for input arguments belonging to the most common domains yet accurate over the full double precision range

sixth order factorization of the evolution operator for time dependent potentials the evolution operator of a quantum system in a time dependent potential is factorized in unitary exponential operators at order 6 this expression is derived with the time ordering method it is compared with lower order factorizations on several simple one dimensional examples better accuracies are reached at sixth order for a given time step than at lower orders due to a significant increase of computation duration per time step the sixth order approximation is mainly useful when high accuracies are required

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

efficient quantum algorithms for simulating sparse hamiltonians we present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse hamiltonian h over a given time t in terms of a procedure for computing the matrix entries of h in particular when h acts on n qubits has at most a constant number of nonzero entries in each row column and h is bounded by a constant we may select any positive integer k such that the simulation requires o log n t1 1 2k accesses to matrix entries of h we also show that the temporal scaling cannot be significantly improved beyond this because sublinear time scaling is not possible

new irrationality measures for q logarithms the three main methods used in diophantine analysis of q series are combined to obtain new upper bounds for irrationality measures of the values of the q logarithm function when p 1 q z 0 1 and z q

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 pressure based algorithm for multi phase flow at all speeds a new finite volume based numerical algorithm for predicting incompressible and compressible multi phase flow phenomena is presented the technique is equally applicable in the subsonic transonic and supersonic regimes the method is formulated on a non orthogonal coordinate system in collocated primitive variables pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the pressure equation is derived from overall mass conservation the performance of the new method is assessed by solving the following two dimensional two phase flow problems i incompressible turbulent bubbly flow in a pipe ii incompressible turbulent air particle flow in a pipe iii compressible dilute gas solid flow over a flat plate and iv compressible dusty flow in a converging diverging nozzle predictions are shown to be in excellent agreement with published numerical and or experimental data

a trotter suzuki approximation for lie groups with applications to hamiltonian simulation we present a product formula to approximate the exponential of a skew hermitian operator that is a sum of generators of a lie algebra the number of terms in the product depends on the structure factors when the generators have large norm with respect to the dimension of the lie algebra or when the norm of the effective operator resulting from nested commutators is less than the product of the norms the number of terms in the product is significantly less than that obtained from well known results we apply our results to construct product formulas useful for the quantum simulation of some continuous variable and bosonic physical systems including systems whose potential is not quadratic for many of these systems we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local hilbert spaces where such a dimension is usually determined by the energy scale of the problem our results emphasize the power of quantum computers for the simulat

quantum monte carlo methods and general decomposition theory of exponential operators and symplectic integrators abstract the present paper reviews briefly the basic ideas of the quantum monte carlo methods the general decomposition theory of ordered exponentials and symplectic integrators

any order propagation of the nonlinear schrodinger equation we derive an exact propagation scheme for nonlinear schrodinger equations this scheme is entirely analogous to the propagation of linear schrodinger equations we accomplish this by defining a special operator whose algebraic properties ensure the correct propagation as applications we provide a simple proof of a recent conjecture regarding higher order integrators for the gross pitaevskii equation extend it to multicomponent equations and to a new class of integrators

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