generic formalism and discrete variational derivative method for the two dimensional vorticity equation the vorticity equation for two dimensional incompressible viscous flows is formulated within the generic formalism for non equilibrium thermodynamics the laws of conservation of energy and increasing entropy derived from the generic formulation are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method the law of increasing entropy corresponds to the dissipation of enstrophy for the vorticity equation some numerical experiments have been done to examine the usefulness of the proposed method

generalizations of arakawa s jacobian abstract a simple method yields discrete jacobians that obey analogues of the differential properties needed to conserve energy and enstrophy in 2 dimensional flow the method is actually independent of the type of discretization and thus applies to an arbitrary representation in gridpoints finite elements or spectral modes or to any mixture of the three we illustrate the method by deriving simple energy and enstrophy conserving jacobians for an irregular triangular mesh in a closed domain and for a mixed gridpoint and mode representation in a semi infinite channel

variational integrators for nonvariational partial differential equations abstract variational integrators for lagrangian dynamical systems provide a systematic way to derive geometric numerical methods these methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the lagrangian via noether s theorem an inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem even though for a large class of systems this requirement is fulfilled there are many interesting examples which do not belong to this class e g equations of advection diffusion type frequently encountered in fluid dynamics or plasma physics on the other hand it is always possible to embed an arbitrary dynamical system into a larger lagrangian system using the method of formal or adjoint lagrangians we investigate the application of the variational integrator method to formal lagrangians and thereby extend the application domain of variational integrators to include potentially all dynamical systems the theory is supported by physically relevant examples such as the advection equation and the vorticity equation and numerically verified remarkably the integrator for the vorticity equation combines arakawa s discretisation of the poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved in the presentation of the results we try to make the geometric framework of variational integrators accessible to non specialists

bracket formulations and energy and helicity preserving numerical methods for the three dimensional vorticity equation abstract the vorticity equation for three dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the poisson or nambu bracket together with a dissipative bracket the budgets of kinetic energy helicity and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method in particular energy and helicity are conserved precisely in inviscid flow computations the energy and enstrophy dissipate properly owing to viscosity in viscous flow computations and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations the relationships between the stream function velocity and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

design and analysis of a dissipative scheme to solve a generalized multi dimensional higgs boson equation in the de sitter space time abstract in this work we design a numerically efficient finite difference technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with riesz fractional derivatives of orders in 0 1 1 2 which considers a generalized potential and a time dependent diffusion factor an energy integral for the mathematical model is readily available and we propose an explicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties as the continuous model a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the 3 1 dimensional de sitter space time and find results qualitatively similar to those available in the literature the present work is the first paper to report on a hamiltonian discretization of the higgs boson equation both fractional and non fractional in the de sitter space time and its numerical analysis more precisely the present manuscript is the first paper of the literature in which a dissipation preserving scheme to solve the multi dimensional fractional higgs boson equation in the de sitter space time is proposed and thoroughly analyzed indeed it is worth pointing out that previous efforts used techniques based on the runge kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed

a new energy and momentum conserving algorithm for the non linear dynamics of shells a numerical time integration scheme for the dynamics of non linear elastic shells is presented that simultaneously and independent of the time step size inherits exactly the conservation laws of total linear total angular momentum as well as total energy the proposed technique generalizes to non linear shells recent work of the authors on non linear elastodynamics and is ideally suited for long term large scale simulations the algorithm is second order accurate and can be immediately extended with no modification to a fourth order accurate scheme the property of exact energy conservation induces a strong notion of non linear numerical stability which manifests itself in actual simulations

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

from finite covariance windows to modeling filters a convex optimization approach the trigonometric moment problem is a classical moment problem with numerous applications in mathematics physics and engineering the rational covariance extension problem is a constrained version of this problem with the constraints arising from the physical realizability of the corresponding solutions although the maximum entropy method gives one well known solution in several applications a wider class of solutions is desired in a seminal paper georgiou derived an existence result for a broad class of models in this paper we review the history of this problem going back to carath eodory as well as applications to stochastic systems and signal processing in particular we present a convex optimization problem for solving the rational covariance extension problem with degree constraint given a partial covariance sequence and the desired zeros of the shaping filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s existence result and his conjecture a generalized version of which we have recently resolved using geometric methods we also survey recent related results on constrained nevanlinna pick interpolation in the context of a variational formulation of the general moment problem

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

on the solution of a generalized higgs boson equation in the de sitter space time through an efficient and hamiltonian scheme abstract the present work is the first paper in the literature to report on a hamiltonian discretization of the fractional higgs boson equation in the de sitter space time and its theoretical analysis more precisely we design herein a numerically efficient finite difference hamiltonian technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with generalized potential and riesz space fractional derivatives of orders in 1 2 an energy integral for the model is readily available and we propose a nonlinear implicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties in the discrete scenario a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the de sitter space time and find results qualitatively similar to those available in the literature for the sake of convenience we provide the matlab code of an alternative linear discretization of the method presented in this work this linear implicit approach is thoroughly analyzed also

integrating factors adjoint equations and lagrangians integrating factors and adjoint equations are determined for linear and non linear differential equations of an arbitrary order the new concept of an adjoint equation is used for construction of a lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables the method is illustrated by considering several equations traditionally regarded as equations without lagrangians noether s theorem is applied to the maxwell equations

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

the cost of complexity in system identification frequency function estimation of finite impulse response systems in this paper we consider full order modeling i e when the true system belongs to the model set we investigate the minimum amount of input energy required to estimate a given linear system with a full order model within a prescribed degree of accuracy as a function of the model complexity this quantity we define to be the cost of complexity the degree of accuracy is measured by the inverse of the maximum variance of the discrete time frequency function estimator over a given frequency range b b it is commonly believed that the cost increases as the model complexity increases however the amount of information that is to be extracted from the system also influences the cost the objective of this paper is to quantify these dependencies for systems described by finite impulse response models it is shown that asymptotically in the model order n and sample size the cost is well approximated by o2n b where o2 is the noise variance this expression can be used as a simple rule of thumb for assessing trade offs that have to be made in a system identification project where full order models are used for example for given experiment duration excitation level and desired accuracy one can assess how the achievable frequency range depends on the required model order this type of consideration is useful when formally planning experiments in addition we establish several properties of the cost of complexity we find for example that if b is very close but not necessarily equal to the optimal input satisfies the model quality constraint for all frequencies

a note on generalized fibonacci sequences consider the generalized fibonacci sequence qn n 0 having initial conditions q0 0 q1 1 and recurrence relation qn aqn 1 qn 2 when n is even or qn bqn 1 qn 2 when n is odd where a and b are nonzero real numbers these sequences arise in a natural way in the study of continued fractions of quadratic irrationals and combinatorics on words or dynamical system theory some well known sequences are special cases of this generalization the fibonacci sequence is a special case of qn with a b 1 pell s sequence is qn with a b 2 and the k fibonacci sequence is qn with a b k in this article we study numerous new properties of these sequences and investigate a sequence closely related to these sequences which can be regarded as a generalization of lucas sequence of the first kind

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

hamiltonian moving particle semi implicit hmps method for incompressible fluid flows particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles and thus have a great degree of flexibility for instance in dealing with the complex motion of surfaces or boundaries in this paper a particle method is developed from a direct discretization of the lagrangian for inviscid incompressible fluid flows the discretized lagrangian is transformed into a hamiltonian with holonomic constraints that stem from the incompressibility condition the rattle algorithm that is a symplectic scheme for holonomically constrained hamiltonian systems is adopted so that the resulting particle method inherits structures possessed by the partial differential equations that govern inviscid incompressible flows the algorithm of the particle method is semi implicit so we call it the hamiltonian moving particle semi implicit hmps method some numerical tests indicate the excellence of the hmps method in conservation of mechanical energy as well as linear and angular momenta

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

input design via lmis admitting frequency wise model specifications in confidence regions a framework for reformulating input design problems in prediction error identification as convex optimization problems is presented for linear time invariant single input single output systems this framework unifies and extends existing results on open loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided a wide range of model quality constraints can be handled in particular different frequency by frequency constraints can be used this opens up new applications of input design in areas such as robust control furthermore quality specifications can be imposed on all models in a confidence region thus allowing for statements such as with at least 99 probability the model quality specifications will be satisfied

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

a hamiltonian particle method for non linear elastodynamics particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles they have a great degree of flexibility for instance in dealing with complex large deformations or the fragmentation of solids in this paper a particle method for non linear elastodynamics of compressible and incompressible materials is developed based on a discretization of the lagrangian from which the governing equations of elastodynamics are derived using the principle of least action the discretized lagrangian leads to a finite dimensional hamiltonian system via the legendre transformation if the material is incompressible the hamiltonian system is accompanied by holonomic constraints depending on whether the material is compressible or incompressible the symplectic scheme adopted for numerical time integration is either the stormer verlet scheme or the rattle method respectively the resulting particle method inherits the symplectic structure possessed by the governing equations of elastodynamics in the case of incompressible materials incompressibility is strictly enforced at each time step some numerical tests indicate the excellence of the method for conservation of mechanical energy besides that of linear and angular momenta

canonical correlation analysis approximate covariance extension and identification of stationary time series abstract in this paper we analyze a class of state space identification algorithms for time series based on canonical correlation analysis in the light of recent results on stochastic systems theory in principle these so called subspace methods can be described as covariance estimation followed by stochastic realization the methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional arma models identification into the solution of a riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity the reason for this is that an essential part of the problem is equivalent to the well known rational covariance extension problem therefore the usual deterministic arguments based on factorization of a hankel matrix are not valid for generic data something that is habitually overlooked in the literature we demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension unless some rather stringent assumptions are made which in general are not explicitly reported in this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory we review the basic ideas of stochastic realization theory in the context of identification discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation the model reduction method of desai and pal 1982 a realization approach to stochastic model reduction proc 1st decision and control conf pp 1105 1112 based on truncated balanced stochastic realizations is partially justified showing that the reduced system structure has a positive covariance sequence but is in general not balanced as a byproduct of this analysis we obtain a theorem prescribing conditions under which the subspace identification methods produce bona fide stochastic systems