the algebraic entropy of classical mechanics we describe the lie algebra of classical mechanics modelled on the lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical poisson bracket it is a polynomially graded lie algebra a class we introduce we describe these lie algebras give an algorithm to calculate the dimensions cn of the homogeneous subspaces of the lie algebra of classical mechanics and determine the value of its entropy limn cn1 n it is 1 825 423 774 201 08 a fundamental constant associated with classical mechanics

physics of symplectic integrators perihelion advances and symplectic corrector algorithms symplectic integrators evolve dynamical systems according to modified hamiltonians whose error terms are also well defined hamiltonians the error of the algorithm is the sum of each error hamiltonian s perturbation on the exact solution when symplectic integrators are applied to the kepler problem these error terms cause the orbit to precess in this work by developing a general method of computing the perihelion advance via the laplace runge lenz vector even for nonseparable hamiltonians i show that the precession error in symplectic integrators can be computed analytically it is found that at leading order each paired error hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period hence symplectic corrector or process integrators which have equal coefficients for these paired error terms will have their precession errors cancel at that order after each period with the use of correctable algorithms both the energy and precession error are of effective order n 2 where n is the nominal order of the algorithm thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions

efficient symplectic runge kutta methods the present paper continues the research in h y liu g sun symplectic rk methods and symplectic prk methods with real eigenvalues j comput math 22 5 2004 769 776 on symplectic runge kutta rk methods with real eigenvalues in a general setting a new but simple proof of the main result in h y liu g sun symplectic rk methods and symplectic prk methods with real eigenvalues j comput math 22 5 2004 769 776 is given that an s stage pth order such method must have that p

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

an assessment of the differential quadrature time integration scheme for nonlinear dynamic equations in 1996 xie an assessment of time integration schemes for non linear dynamic equations journal of sound and vibration 192 1 1996 321 331 presented an assessment on seven existing and commonly used time integration schemes for nonlinear dynamic equations in this work the differential quadrature dq time integration scheme proposed by fung in 2001 is assessed following the same procedures as xie s it is shown that accurate numerical results can be obtained by the dq method when using much larger time step over the commonly used time integration schemes based on the results reported herein some conclusions are drawn

the split decomposition of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering v0 v1 vd of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 vd 1 0 iii there exists an ordering v 0 v 1 v of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 v 1 0 iv there is no subspace w of v such that both aw w a w w other than w 0 and w v we call such a pair a tridiagonal pair on v in this note we obtain two results first we show that each of a a is determined up to affine transformation by the vi and v i secondly we characterize the case in which the vi and v i all have dimension one we prove both results using a certain decomposition of v called the split decomposition

symmetric second derivative integration methods abstract the purpose of this paper is the derivation of symmetric second derivative general linear methods sglms for general time reversible differential equations for this purpose we find algebraic conditions on the coefficients matrices of the methods which are sufficient for the method to be symmetric some symmetric methods of order 4 are constructed and implemented on the famous reversible problems numerical experiments show that the constructed symmetric sglms approximately conserve the invariants of motion over long time intervals for reversible hamiltonian systems

a fully adaptive reaction diffusion integration scheme with applications to systems biology adaptive integration schemes for ode systems typically function by adjusting the time step size so as to keep the truncation error below some desired value for adaptive integration of pde systems involving coupled kinetic reaction and diffusion operations truncation error arises not only from the individual propagators but also from their method of coupling a common second order accurate method for coupling operators is strang s method of operator splitting we derive an expression for the truncation error resulting from strang splitting reaction and diffusion operators for an arbitrary number of spatial dimensions and demonstrate its use in adaptive time step algorithms in addition we present explanations of the second order implicit reaction and diffusion operators and their individual error calculations used in our implementation of the scheme finally using example simulations we discuss the use of this calculation for problems in systems biology

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

numerical integrators for the hybrid monte carlo method we construct numerical integrators for hamiltonian problems that may advantageously replace the standard verlet time stepper within hybrid monte carlo and related simulations past attempts have often aimed at boosting the order of accuracy of the integrator and or reducing the size of its error constants order and error constant are relevant concepts in the limit of vanishing step length we propose an alternative methodology based on the performance of the integrator when sampling from gaussian distributions with not necessarily small step lengths we construct new splitting formulae that require two three or four force evaluations per time step limited proof of concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard verlet method especially in problems with high dimensionality

the generalized terwilliger algebra and its finite dimensional modules when d 2 terwilliger j algebraic combin 1 1992 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations are often called the triple product relations they determine much of the structure of t though not all of it in general to illuminate the role these relations play the current author introduced j algebra233 2000 213 252 a generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c the dimensions of c and c are equal let d 1 denote this common dimension intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s triple product relations t is infinite dimensional and noncommutative in general in this paper we study t and its finite dimensional modules when d 2 and t has no extra vanishing intersection numbers or dual intersection numbers in this case we show t is c algebra isomorphic to m3 c a where m3 c denotes the c algebra consisting of all 3 by 3 matrices with entries in c and a denotes the associative c algebra with identity generated by the symbols e and f subject to the relations e2 e and f2 f we find a basis for a and we determine the center of a we classify the finite dimensional indecomposable a modules up to isomorphism there are four such a modules in every odd dimension and in every even dimension these modules are parameterized by a single complex number we also classify the finite dimensional irreducible a modules up to isomorphism using our results concerning a we find a basis for t we describe the center of t and we classify both the finite dimensional indecomposable and the finite dimensional irreducible t modules up to isomorphism

an assessment of time integration schemes for non linear dynamic equations abstract in the transient analysis of engineering structures by the finite element method there are several commonly used time integration schemes such as the method of central difference the newmark method the wilson method the houbolt method and the method of hilber hughes taylor in this paper these time integration schemes are applied to simple non linear dynamic systems and the performances of these schemes are compared it is found that there are significant differences in the accuracy of these schemes the result of some of these schemes are so inaccurate that it becomes clear that they are not suited for time integration over a long time duration the behaviour of these schemes is also compared with that of the fourth order runge kutta method which has been the preferred integration scheme for non linear systems of a single or a few degrees of freedom

high order symplectic runge kutta nystro m methods a numerical method for ordinary differential equations is called symplectic if when applied to hamiltonian problems it preserves the symplectic structure in phase space thus reproducing the main qualitative property of solutions of hamiltonian systems the authors construct and test symplectic explicit runge kutta nystrom rkn methods of order 8 the outcome of the investigation is that existing high order symplectic rkn formulae require so many evaluations per step that they are much less efficient than conventional eighth order nonsymplectic variable step size integrators even for low accuracy however symplectic integration is of use in the study of qualitative features of the systems being integrated

composite methods for numerical solution of stiff systems of ode s this paper is concerned with methods which consist of sequential application of several different schemes for stiff ordinary differential systems with a predetermined ratio of step lengths we show that under some conditions such composite methods have higher order and better stability than the constituent schemes compositions of obrechkoff methods and of implicit runge kutta processes are investigated

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

an almost symmetric strang splitting scheme for the construction of high order composition methods in this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the partial flows that results from the splitting procedure cannot be computed exactly instead we insert a well chosen state y into the corresponding nonlinearity b y y which results in a linear term b y y whose exact flow can be determined efficiently therefore in the spirit of splitting methods it is still possible for the numerical simulation to satisfy certain properties of the exact flow however strang splitting is no longer symmetric even though it is still a second order method and thus high order composition methods are not easily attainable we will show that an iterated strang splitting scheme can be constructed which yields a method that is symmetric up to a given order this method can then be used to attain high order composition schemes we will illustrate our theoretical results up to order six by conducting numerical experiments for a charged particle in an inhomogeneous electric field a post newtonian computation in celestial mechanics and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to strang splitting for the first example we also perform a comparison with the standard fourth order runge kutta methods and find significant gains in efficiency as well better conservation properties

on higher order semi explicit symplectic partitioned runge kutta methods for constrained hamiltonian systems in this paper we generalize the class of explicit partitioned runge kutta prk methods for separable hamiltonian systems to systems with holonomic constraints for a convenient analysis of such schemes we first generalize the backward error analysis for systems in bbb r m to systems on manifolds embedded in bbb r m by applying this analysis to constrained prk methods we prove that such methods will in general suffer from order reduction as well known for higher index differential algebraic equations however this order reduction can be avoided by a proper modification of the standard prk methods this modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant we also give a numerical comparison of several second fourth and sixth order methods

symplectic partitioned runge kutta methods with the phase lag property abstract in this work specially tuned symplectic partitioned runge kutta sprk methods with minimum phase lag and phase fitted have been considered the general framework for constructing sprk methods with minimum phase lag is given

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

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

a refinement of the split decomposition of a tridiagonal pair let v denote a nonzero finite dimensional vector space over a field k and let a a denote a tridiagonal pair on v of diameter d let v u0 ud denote the split decomposition and let i denote the dimension of ui in this paper at first we show there exists a unique integer h 0 h d 2 such that i 1 i for d h i d we call h the height of the tridiagonal pair for 0 r h we define subspaces ui r r i d r by ui r ri r ur kerrd 2r 1 where r denotes the rasing map we show v is decomposed as a direct sum v r 0h i rd rui r this gives a refinement of the split decomposition define u r i rd rui r and observe v r 0hu r we show lu r u r 1 u r u r 1 for 0 r h where we set u 1 u h 1 0 let f r v u r denote the projection we show the lowering map l is decomposed as l l l 0 l where l r 1hf r 1 lf r l 0 r 0hf r lf r and l r 0h 1f r 1 lf r these maps satisfy l u r u r 1 l 0 u r u r and l u r u r 1 for 0 r h the main results of this paper are the following i for 0 r h 1 and r 2 i d r 1 rl l r holds on ui r for some scalar ii for 1 r h and r i d r 1 rl l r holds on ui r for some scalar iii for 0 r h and r 1 i d r 1 rl 0 l 0 r i holds on ui r for some scalars moreover we give explicit expressions of

spurious behavior of a symplectic integrator in this article we study the existence and behavior of spurious solutions of symplectic euler method for some hamiltonian systems it is shown that the symplectic integrator applied to hamiltonian systems in general doesn t avoid spurious behavior even spurious period two solutions the numerical results are presented

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

buckling and post buckling analysis of extensible beam columns by using the differential quadrature method buckling and post buckling analysis of extensible beam columns is performed numerically in this paper it was experienced earlier that in some cases the numerical integration would not produce the convergent post buckling solutions especially under high loads therefore a new differential quadrature dq based iterative numerical integration method is proposed to solve post buckling differential equations of extensible beam columns six cases including five classical euler buckling cases are analyzed critical loads and convergent post buckling solutions under different applied loads are obtained the results are compared with the existing multiple scales solutions it is found that under high applied loads the small rotation assumption in obtaining multiple scales solutions is no longer valid the proposed iterative dq based numerical integration method can yield reliable and accurate post buckling solutions even at high applied loads for the cases investigated

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

composite integrators for bi hamiltonian systems abstract symmetric composition methods are applied to dynamical systems in bi hamiltonian form to lotka volterra equations to the completely integrable lorenz equation and to the periodic toda lattice the numerical results obtained show that the hamiltonians are preserved with high accuracy and the periodicity of the solutions are retained

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

on the terwilliger algebra of bipartite distance regular graphs with δ2 0 and c2 1 abstract let denote a bipartite distance regular graph with diameter d 4 and valency k 3 let x denote the vertex set of and let a denote the adjacency matrix of for x x and for 0 i d let i x denote the set of vertices in x that are distance i from vertex x define a parameter 2 in terms of the intersection numbers by 2 k 2 c 3 1 c 2 1 p 22 2 we first show that 2 0 implies that d 5 or c 2 1 2 for x x let t t x denote the subalgebra of mat x c generated by a e 0 e 1 e d where for 0 i d e i represents the projection onto the ith subconstituent of with respect to x we refer to t as the terwilliger algebra of with respect to x by the endpoint of an irreducible t module w we mean min i e i w 0 in this paper we assume has the property that for 2 i d 1 there exist complex scalars i i such that for all x y z x with x y 2 x z i y z i we have i i 1 x 1 y i 1 z i 1 x i 1 y 1 z we additionally assume that 2 0 with c 2 1 under the above assumptions we study the algebra t we show that if is not almost 2 homogeneous then up to isomorphism there exists exactly one irreducible t module with endpoint 2 we give an orthogonal basis for this t module and we give the action of a on this basis

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

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

multi revolution composition methods for highly oscillatory differential equations we introduce a new class of multi revolution composition methods for the approximation of the n n th iterate of a given near identity map when applied to the numerical integration of highly oscillatory systems of differential equations the technique benefits from the properties of standard composition methods it is intrinsically geometric and well suited for hamiltonian or divergence free equations for instance we prove error estimates with error constants that are independent of the oscillatory frequency numerical experiments in particular for the nonlinear schrodinger equation illustrate the theoretical results as well as the efficiency and versatility of the methods

accurate and efficient simulation of rigid body rotations abstract this paper introduces efficient and accurate algorithms for simulating the rotation of a three dimensional rigid object and compares them to several prior methods first we introduce a second order accurate method that incorporates a third order correction then we introduce a third order accurate method and finally we give a fourth order accurate method these methods are single step and the update operation is only a single rotation the algorithms are derived in a general lie group setting second we introduce a near optimal energy correction method which allows exact conservation of energy this algorithm is faster and easier to implement than implicit methods for exact energy conservation our third order method with energy conservation is experimentally seen to act better than a fourth order accurate method these new methods are superior to naive runge kutta or predictor corrector methods which are only second order accurate for sphere valued functions the second order symplectic mclachlan reich methods are observed to be excellent at approximate energy conservation but are not as good at long term accuracy as our best methods finally we present comparisons with fourth order accurate symplectic methods which have good accuracy but higher computational cost

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

the terwilliger algebra of a distance regular graph of negative type abstract let denote a distance regular graph with diameter d 3 assume has classical parameters d b with b 1 let x denote the vertex set of and let a mat x c denote the adjacency matrix of fix x x and let a mat x c denote the corresponding dual adjacency matrix let t denote the subalgebra of mat x c generated by a a we call t the terwilliger algebra of with respect to x we show that up to isomorphism there exist exactly two irreducible t modules with endpoint 1 their dimensions are d and 2 d 2 for these t modules we display a basis consisting of eigenvectors for a and for each basis we give the action of a

preservation of stability properties near fixed points of linear hamiltonian systems by symplectic integrators abstract based on reasonable testing model problems we study the preservation by symplectic runge kutta method srk and symplectic partitioned runge kutta method sprk of structures for fixed points of linear hamiltonian systems the structure preservation region provides a practical criterion for choosing step size in symplectic computation examples are given to justify the investigation

solving initial value problems by differential quadrature method part 1 first order equations in this paper the differential quadrature method is used to solve first order initial value problems the initial condition is given at the beginning of a time interval the time derivative at a sampling grid point within the time interval can be expressed as a weighted linear sum of the given initial condition and the function values at the sampling grid points within the time interval the order of accuracy and the stability property of the quadrature solutions depend on the locations of the sampling grid points it is shown that the order of accuracy of the quadrature solutions at the end of a time interval can be improved to 2n 1 or 2n if the n sampling grid points are chosen carefully in fact the approximate solutions are equivalent to the generalized pade approximations the resultant algorithms are therefore unconditionally stable with controllable numerical dissipation the corresponding sampling grid points are found to be given by the roots of the modified shifted legendre polynomials from the numerical examples the accuracy of the quadrature solutions obtained by using the proposed sampling grid points is found to be better than those obtained by the commonly used uniformly spaced or chebyshev gauss lobatto sampling grid points copyright 2001 john wiley sons ltd

explicit unitary schemes to solve quantum operator equations of motion to solve operator equations of motion in quantum mechanics and in a quantum scalar field theory we propose an explicit unitary scheme with arbitrary high order of accuracy

the time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems hence the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging in this work we show that time finite element methods because of their inherent energy momentum conserving property in the case of linear and nonlinear elastodynamics provide a robust time stepping method for nonlinear dynamic equations including chaotic systems we also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework thus the time finite element provides a unification of time stepping schemes used in diverse disciplines we demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic

time integration methods for finite element discretisations of the second order maxwell equation this article deals with time integration for the second order maxwell equations with possibly non zero conductivity in the context of the discontinuous galerkin finite element method dg fem and the h curl conforming fem for the spatial discretisation hierarchic h curl conforming basis functions are used up to polynomial order p 3 over tetrahedral meshes meaning fourth order convergence rate a high order polynomial basis often warrants the use of high order time integration schemes but many well known high order schemes may suffer from a severe time step stability restriction owing to the conductivity term we investigate several possible time integration methods from the point of view of accuracy stability and computational work we also carry out a numerical fourier analysis to study the dispersion and dissipation properties of the semi discrete dg fem scheme as well as the fully discrete schemes with several of the time integration methods the dispersion and dissipation properties of the spatial discretisation and those of the time integration methods are investigated separately providing additional insight into the two discretisation steps

a unified approach to spurious solutions introduced by time discretization part i basic theory the asymptotic states of numerical methods for initial value problems are examined in particular spurious steady solutions solutions with period 2 in the timestep and spurious invariant curves

high order staggered schemes for compressible hydrodynamics weak consistency and numerical validation abstract staggered grids schemes formulated in internal energy are commonly used for cfd applications in industrial context here we prove the consistency of a class of high order lagrange remap staggered schemes for solving the euler equations in 1d and 2d on cartesian grids the main result of the paper is that using an a posteriori internal energy corrector the lagrangian schemes are proved to be conservative in mass momentum and total energy and to be weakly consistent with the 1d lagrangian formulation of the euler equations extension in 2d is done using directional splitting methods and face staggering numerical examples in both 1d and 2d illustrate the accuracy the convergence and the robustness of the schemes

splitting methods for non autonomous separable dynamical systems a large number of splitting methods for autonomous separable systems exist in the literature which have been designed for many different structures of the vector field however the performance of most of these methods is diminished and their orders of accuracy are frequently reduced when applied to non autonomous problems based on the formal solution obtained from the magnus series expansion we show how to modify a standard splitting method for autonomous problems to treat non autonomous systems with similar or better efficiency we illustrate this technique to build new fourth and sixth order schemes whose performance is then illustrated on several numerical examples

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

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

statistical properties of the energy in time dependent homogeneous power law potentials we study classical 1d hamilton systems with a homogeneous power law potential and their statistical behavior assuming a microcanonical distribution of the initial conditions and describing its change under a monotonically increasing time dependent function a t the prefactor of the potential using the nonlinear wentzel kramers brillouin like method of papamikos and robnik 2012 j phys a math theor 44 315102 and following a previous work by papamikos and robnik 2011 j phys a math theor 45 015206 we specifically analyze the mean energy the variance and the adiabatic invariant action of the systems for large time and we show that the mean energy and variance increase as powers of a t while the action oscillates and finally remains constant by means of a number of detailed case studies we show that the theoretical prediction is excellent which demonstrates the usefulness of the method in such applications

numerical dynamics of a nonstandard finite difference method for a class of delay differential equations abstract this paper proposes a nonstandard finite difference nsfd method for a class of delay differential equations we prove that for any step size h 1 m m z this numerical method could intrinsically preserve the qualitative behavior of the dynamical system including the local stability of equilibrium the existence and the direction of hopf bifurcation and the stability of bifurcating periodic solution finally to illustrate the analytic results the nsfd method is used for a model of the survival of red blood cells in animals

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

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

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

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

splitting methods for non autonomous hamiltonian equations abstract we present an algorithm for numerically integrating non autonomous hamiltonian differential equations special attention is paid to the separable case and in particular a new fourth order splitting method is presented which in a certain measure is optimal in combination with a new way of handling non autonomous problems the schemes we present are based on magnus expansions and they show very promising results when applied to hamiltonian odes and pdes

multisymplectic schemes for strongly coupled schrodinger system in this paper two semi explicit multisymplectic schemes are derived for the strongly coupled schrodinger system based on the two new multisymplectic schemes we obtain a multisymplectic composition scheme which improves the accuracy in time the best merits of the present schemes are all implemented easily some numerical simulations are done for investigating nonlinear coupling and linear coupling numerical results indicate that the new multisymplectic composition scheme is effective