interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

interference alignment with asymmetric complex signaling settling the host madsen nosratinia conjecture it has been conjectured by ho madsen and nosratinia that complex gaussian interference channels with constant channel coefficients have only one degree of freedom regardless of the number of users while several examples are known of constant channels that achieve more than 1 degree of freedom these special cases only span a subset of measure zero in other words for almost all channel coefficient values it is not known if more than 1 degree of freedom is achievable in this paper we settle the host madsen nosratinia conjecture in the negative we show that at least 1 2 degrees of freedom are achievable for all values of complex channel coefficients except for a subset of measure zero for the class of linear beamforming and interference alignment schemes considered in this paper it is also shown that 1 2 is the maximum number of degrees of freedom achievable on the complex gaussian 3 user interference channel with constant channel coefficients for almost all values of channel coefficients to establish the achievability of 1 2 degrees of freedom we use the novel idea of asymmetric complex signaling i e the inputs are chosen to be complex but not circularly symmetric it is shown that unlike gaussian point to point multiple access and broadcast channels where circularly symmetric complex gaussian inputs are optimal for interference channels optimal inputs are in general asymmetric with asymmetric complex signaling we also show that the 2 user complex gaussian x channel with constant channel coefficients achieves the outer bound of 4 3 degrees of freedom i e the assumption of time variations frequency selectivity used in prior work to establish the same result is not needed

generalized degrees of freedom of the symmetric gaussian k user interference channel we characterize the generalized degrees of freedom of the k user symmetric gaussian interference channel where all desired links have the same signal to noise ratio snr and all undesired links carrying interference have the same interference to noise ratio inr snr we find that the number of generalized degrees of freedom per user d does not depend on the number of users so that the characterization is identical to the 2 user interference channel with the exception of a singularity at 1 where d 1 1 k the achievable schemes use multilevel coding with a nested lattice structure that opens the possibility that the sum of interfering signals can be decoded at a receiver even though the messages carried by the interfering signals are not decodable

hamiltonian formalism for nonlocal lagrangians a hamiltonian formalism is set up for nonlocal lagrangian systems the method is based on obtaining an equivalent singular first order lagrangian which is processed according to the standard legendre transformation and then the resulting hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints finally the standard results for local lagrangians of any order are obtained as a particular case

gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

estimate of the difference between the kac operator and the schrodinger semigroup an operator norm estimate of the difference between the kac operator and the schrodinger semigroup is proved and used to give a variant of the trotter product formula for schrodinger operators in thelp operator norm this extends helffer s result in thel2 operator norm to the case in thelp operator norm for more general scalar potentials and with vector potentials the method of the proof is probabilistic based on the feynman kac and feynman kac ito formula

splitting methods for constrained diffusion reaction systems abstract we consider lie and strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term since such systems are known to be sensitive with respect to perturbations the splitting procedure seems promising as we can treat the nonlinearity separately this has some computational advantages since we only have to solve a linear constrained system and a nonlinear ordinary differential equation however strang splitting suffers from order reduction which limits its efficiency this reduction is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem the incorporation of an additional correction term resolves this problem without increasing the computational cost significantly numerical examples including a coupled mechanical system illustrate the proved convergence results

dense output for extrapolation methods this paper is concerned with dense output formulas for extrapolation methods for ordinary differential equations in particular the extrapolated explicit euler method the gbs method for non stiff equations and the extrapolated linearly implicit euler method for stiff and differential algebraic equations are considered existence and uniqueness questions for dense output formulas are discussed and an algorithmic description for their construction is given several numerical experiments illustrate the theoretical results

an a posteriori a priori analysis of multiscale operator splitting in this paper we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reaction diffusion equation our goal is to analyze and accurately estimate the error of the numerical solution including the effects of any instabilities that can result from multiscale operator splitting we present both an a priori error analysis and a new type of hybrid a priori a posteriori error analysis for an operator splitting discontinuous galerkin finite element method both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem the hybrid analysis has the form of a computable a posteriori leading order expression and a provably higher order a priori expression the hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways in particular this provides the means to monitor global instabilities that can arise from operator splitting

error analysis of a specialized numerical method for mathematical models from neuroscience abstract the exponential euler method is a nonstandard approximation scheme that was developed specifically for the hodgkin huxley differential equation models that arise in neuroscience and was one of the discretization schemes used in the neural systems package called genesis in this article we show the scheme is first order accurate develop a second order accurate extension and suggest ways the method can be used to compute approximations to certain time dependent partial differential equations we also apply the scheme to an integro differential equation model of neuronal activity and furnish sample computations that illustrate our theoretical results

numerical solution of implicit neutral functional differential equations this paper is concerned with the numerical solution of implicit neutral functional differential equations based on the continuous runge kutta method for ordinary differential equations and the collocation method for functional equations two general one step methods are formulated and their uniform order of approximation is discussed numerical stability of a class of runge kutta collocation methods is analyzed in the case where a neutral equation can be expressed in an explicit as well as an implicit form the advantage of constructing a numerical scheme directly from the implicit form is explained theoretically and demonstrated by a numerical example

convergence of the peaceman rachford approximation for reaction diffusion systems consider a reaction diffusion system of the form u t m u f u 0 where m is a m m matrix whose spectrum is included in z 0 we approximate it by the peaceman rachford approximation defined by p t 1 tf 2 1 1 tm 2 1 tm 2 1 1 tf 2 we prove convergence of this scheme and show that it is of order two

a fifth order interpolant for the dornand and prince runge kutta method abstract a family of fifth order interpolants for the fifth order solution provided by the dormand and prince runge kutta pair rk5 4 7m which requires two additional function evaluations per step is presented an optimal interpolant in this family has been determined by choosing the parameters to minimize the leading coefficients of the local truncation error of the continuous solution some numerical experiments with the nonstiff detest problems show that the proposed optimal method has a good interpolatory behavior

an efficient numerical scheme for precise time integration of a diffusion dissolution precipitation chemical system a numerical scheme based on an operator splitting method and a dense output event location algorithm is proposed to integrate a diffusion dissolution precipitation chemical initial boundary value problem with jumping nonlinearities the numerical analysis of the scheme is carried out and it is proved to be of order 2 in time this global order estimate is illustrated numerically on a test case

an estimate of the kac transfer operator abstract let v be a positive potential class c 4 on r n such that its derivatives of order 2 to 4 are bounded we prove that exp tv exp 2t exp tv exp 2t v l l 2 o t 2 as t tends to zero our assumptions are more general than those of helffer 2 and our techniques rely on holomorphic semigroups and estimates on commutators soit v un potentiel positif de classe c 4 sur r n tel que ses derivees d ordre 2 a 4 soient bornees nous prouvons que exp tv exp 2t exp tv exp 2t v l l 2 o t 2 quand t tend vers 0 nos hypotheses sont plus generales que celles de helffer 2 et nous utilisons des techniques de semi groupes holomorphes et des estimations sur des commutateurs

diffusion and dissolution precipitation in an open porous reactive medium in this article a mathematical model is proposed for dissolution precipitation with diffusion in a reactive open porous medium when many chemical species in solid phase can dissolve for this purpose the concept of saturation concentration for a chemical species in liquid phase is introduced and the mathematical properties of the functions representing such saturation concentrations are investigated then the equations of dissolution precipitation are stated and investigated in order to be reformulated as an obstacle problem finally an original mathematical model is derived and this model can be used for predictive numerical simulations

communication over mimo x channels interference alignment decomposition and performance analysis in a multiple antenna system with two transmitters and two receivers a scenario of data communication known as the x channel is studied in which each receiver receives data from both transmitters in this scenario it is assumed that each transmitter is unaware of the other transmitter s data noncooperative scenario this system can be considered as a combination of two broadcast channels from the transmitters points of view and two multiple access channels from the receivers points of view taking advantage of both perspectives two signaling schemes for such a scenario are developed in these schemes some linear filters are employed at the transmitters and at the receivers which decompose the system into either two noninterfering multiple antenna broadcast subchannels or two noninterfering multiple antenna multiple access subchannels the main objective in the design of the filters is to exploit the structure of the channel matrices to achieve the highest multiplexing gain mg it is shown that the proposed noncooperative signaling schemes outperform other known noncooperative schemes in terms of the achievable mg in particular it is shown that in some specific cases the achieved mg is the same as the mg of the system if full cooperation is provided either between the transmitters or between the receivers in the second part of the paper it is shown that by using mixed design schemes rather than decomposition schemes and taking the statistical properties of the interference terms into account the power offset of the system can be improved the power offset represents the horizontal shift in the curve of the sum rate versus the total power in decibels

on the simultaneous numerical integration of an ivp and its associated variational equation abstract in this paper we consider the numerical solution of an ivp together with its variational equation by runge kutta methods using the fact that the jacobian matrix has a particular structure we develop an algorithm based on methods of interpolatory type which performs the numerical integration more efficiently than standard methods and with the possibility of saving jacobian evaluations some numerical experiments are presented

inner bound on the gdof of the k user mimo gaussian symmetric interference channel the k user multiple input multiple output mimo gaussian symmetric interference channel where each transmitter has m antennas and each receiver has n antennas is studied from a generalized degrees of freedom gdof perspective an inner bound on the gdof is derived using a combination of techniques such as treating interference as noise zero forcing zf at the receivers interference alignment ia and extending the han kobayashi hk scheme to k users as a function of the number of antennas and the log inr log snr level several interesting conclusions are drawn from the derived bounds it is shown that when k n m 1 a combination of the hk and ia schemes performs the best among the schemes considered when n m k n m 1 the hk scheme outperforms other schemes and is found to be gdof optimal in many cases in addition when the snr and inr are at the same level zf receiving and the hk scheme have the same gdof performance

outer bounds on the sum rate of the k user mimo gaussian interference channel this paper derives outer bounds on the sum rate of the k user mimo gaussian interference channel gic three outer bounds are derived under different assumptions of cooperation and providing side information to receivers the novelty in the derivation lies in the careful selection of side information which results in the cancellation of the negative differential entropy terms containing signal components leading to a tractable outer bound the overall outer bound is obtained by taking the minimum of the three outer bounds the derived bounds are simplified for the mimo gaussian symmetric ic to obtain outer bounds on the generalized degrees of freedom gdof the relative performance of the bounds yields insight into the performance limits of multiuser mimo gics and the relative merits of different schemes for interference management these insights are confirmed by establishing the optimality of the bounds in specific cases using an inner bound on the gdof derived by the authors in a previous work it is also shown that many of the existing results on the gdof of the gic can be obtained as special cases of the bounds e g by setting k 2 or the number of antennas at each user to 1

quantal symmetry for a system with a singular higher order lagrangian based on the phase space generating functional of the green function for a constrained hamiltonian system with a singular higher order lagrangian the canonical ward identities for such a system under the local and global transformation have been derived respectively the quantal conserved charge qcc under the global symmetry transformation is also deduced in general these qccs are different from the noether charge in classical theory a comparison of these quantal conservation laws with those deriving from the configuration space path integral for gauge invariant theories is discussed we give a preliminary application of our results to yang mills ym theory and chern simons cs theory with higher order derivatives a new form of gauge ghost proper vertices and new conserved charges at the quantum level are obtained for the ym theory the quantal becchi rouet stora brs conserved charges and conserved angular momentum are also derived for cs theory the advantage of our canonical formalism is that we do not carry out the integration over the canonical momenta in the phase space path integral as usually performed

reductions of topologically massive gravity i hamiltonian analysis of second order degenerate lagrangians we present skinner rusk and hamiltonian formalisms of second order degenerate clement and sarioglu tekin lagrangians the dirac bergmann constraint algorithm is employed to obtain hamiltonian realizations of lagrangian theories the gotay nester hinds algorithm is used to investigate skinner rusk formalisms of these systems

higher order operator splitting methods via zassenhaus product formula theory and applications abstract in this paper we contribute higher order operator splitting methods improved by zassenhaus product we apply the contribution to classical and iterative splitting methods the underlying analysis to obtain higher order operator splitting methods is presented while applying the methods to partial differential equations the benefits of balancing time and spatial scales are discussed to accelerate the methods the verification of the improved splitting methods are done with numerical examples an individual handling of each operator with adapted standard higher order time integrators is discussed finally we conclude the higher order operator splitting methods

an efficient scalable numerical algorithm for the simulation of electrochemical systems on irregular domains we present a projection method for the solution of the diffusive transport and reaction equations of electrochemical systems on irregular time dependent domains specific applications include electrodeposition of copper in sub micron trenches as well as any other electrochemical system with an arbitrarily shaped bulk region of dilute electrolyte solution our method uses a finite volume spatial discretization that is second order accurate throughout including a nonuniform region used as a transition to the far field chemical concentrations time integration is performed with a splitting technique that includes a projection step to solve for the electric potential the resulting method is first order accurate in time and is observed to be stable for relatively large time steps furthermore the algorithm complexity scales very respectably with grid refinement and is naturally parallelizable

reductions of topologically massive gravity ii first order realizations of second order lagrangians second order degenerate clement and sarioglu tekin lagrangians are casted into forms of various first order lagrangians the structures of the iterated tangent bundle and acceleration bundle are presented as a suitable geometric framework hamiltonian analyses of these equivalent formalisms are performed by means of the dirac bergmann constraint algorithm all formulations are shown to possess only second class constraints

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

derivation of many body potential among charged particles in the s matrix method a general method of deriving a classical potential from the s matrix element of particle scattering in the theory of quantized fields is applied to electrodynamics to the post post coulombian approximation to obtain the many body potential a consistent prescription is implemented in subtracting the contributions of the repetition of lower order potential from the s matrix elements of the higher order diagrams the result shows that the four body potential between charged particles has a characteristic feature at a large distance and the two body potential is identical with that given in the reduced hamiltonian of wheeler feynman electrodynamics the advantage of the s matrix method over the canonical formalism is to give the potential directly without complicated treatment of the interaction with higher derivatives by a method of constrained dynamics

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

analysis of splitting methods for reaction diffusion problems using stochastic calculus we consider linear and nonlinear reaction diffusion problems and their time dis cretization by splitting methods we give probabilistic interpretations of the splitting schemes and show how these representations allow to give error bounds for the deter ministic propagator under weak hypothesis on the reaction part to show these results we only use the it o formula and basic properties of solutions of stochastic differential equations eventually we show how probabilistic representations of splitting schemes can be used to derive hybrid numerical schemes based on monte carlo approxima tions of the splitting method itself

solving optimization constrained differential equations with discontinuity points with application to atmospheric chemistry ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time a particular application arises in the modeling of atmospheric particles discontinuity points are created by the activation deactivation of inequality constraints a numerical method for the solution of optimization constrained differential equations is proposed by coupling an implicit runge kutta method radau5 with numerical techniques for the detection of the events activation and deactivation of constraints the computation of the events is based on dense output formulas continuation techniques and geometric arguments numerical results are presented for the simulation of the time dependent equilibrium of organic atmospheric aerosol particles and show the efficiency and accuracy of the approach

an efficient runge kutta 4 5 pair abstract a pair of explicit runge kutta formulas of orders 4 and 5 is derived it is significantly more efficient than the fehlberg and dormand prince pairs and by standard measures it is of at least as high quality there are two independent estimates of the local error the local error of the interpolant is to leading order a problem independent function of the local error at the end of the step

multistep high order interpolants of runge kutta methods abstract we consider a p order runge kutta method k n i x n c i h y n h j 1 a ij k n j i 1 y n 1 y n h i 1 b i k n i for solving an initial value problem for ordinary differential equations the aim of this paper is to construct p order interpolants by using the values furnished by the method on n successive intervals of integration by using lagrange interpolation one can obtain a p order interpolant over p intervals but we are interested in finding the minimum number of intervals needed to obtain this we provide the conditions to be satisfied and we obtain an estimation of the number n some examples are given

treatment of higher order lagrangians via the construction of dynamically equivalent first order lagrangians for a given in general singular lagrangian containing higher order time derivatives a dynamically equivalent lagrangian with only first order time derivatives is constructed a hamiltonian structure for this first order lagrangian is then found with the use of the dirac theory of constraints it is shown that in the case of a nonsingular higher order lagrangian the ostrogradsky dynamics is derived in this way further it is shown that ambiguities characteristic of higher order lagrangian systems do not appear when using this construction in particular it is shown that the addition of a total time derivative term to the higher order lagrangian can only induce a time independent canonical transformation even in the case of a singular lagrangian

efficient boundary corrected strang splitting abstract strang splitting is a well established tool for the numerical integration of evolution equations it allows the application of tailored integrators for different parts of the vector field however it is also prone to order reduction in the case of non trivial boundary conditions this order reduction can be remedied by correcting the boundary values of the intermediate splitting step in this paper three different approaches for constructing such a correction in the case of inhomogeneous dirichlet neumann and mixed boundary conditions are presented numerical examples that illustrate the effectiveness and benefits of these corrections are included

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

a new embedded pair of runge kutta formulas of orders 5 and 6 a new pair of embedded runge kutta rk formulas of orders 5 and 6 is presented it is derived from a family of rk methods depending on eight parameters by using certain measures of accuracy and stability numerical tests comparing its efficiency to other formulas of the same order in current use are presented with an extra function evaluation per step a c 1 continuous interpolant of order 5 can be obtained

the behaviour of the local error in splitting methods applied to stiff problems splitting methods are frequently used in solving stiff differential equations and it is common to split the system of equations into a stiff and a nonstiff part the classical theory for the local order of consistency is valid only for stepsizes which are smaller than what one would typically prefer to use in the integration error control and stepsize selection devices based on classical local order theory may lead to unstable error behaviour and inefficient stepsize sequences here the behaviour of the local error in the strang and godunov splitting methods is explained by using two different tools lie series and singular perturbation theory the two approaches provide an understanding of the phenomena from different points of view but both are consistent with what is observed in numerical experiments

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

convergence of a splitting method of high order for reaction diffusion systems in this article we prove the convergence of a splitting scheme of high order for a reaction diffusion system of the form u t m u f u 0 where m is an m m matrix whose spectrum is included in rz 0 this scheme is obtained by applying a richardson extrapolation to a strang formula

a dynamical formalism of singular lagrangian system with higher derivatives the singular lagrangian system with higher derivatives is analyzed with the aid of the ostrogradski transformation and the dirac formalism the formulation of canonical theory is developed so that the equivalence between the lagrange formalism and the hamilton one is maintained as a practical example the acceleration dependent potentials appearing in the lagrangian of two point particles interacting gravitationally are dealt with and the equivalence between the two hamiltonians that follow from the two lagrangians which are related by the coordinate transformations is shown it is also shown when the constraints are all first class that a consistent generator of gauge transformation is constructed typical examples are given

successive interference mitigation in multiuser mimo channels motivated by the work of dahrouj and yu in applying the han kobayashi transmission strategy for mitigating the intercell interference in a multi cell multi user multiple input single output interference network miso in this paper considers splitting messages into private and common parts in a multi cell multi user mimo in specifically the covariances of the private messages and common messages are designed to optimize either the sum rate or the minimal rate the common messages and private messages are decoded in sequence using successive decoding this paper shows how these difficult optimization problems can be adequately solved by means of d c d ifference of c oncave functions optimization over a simple convex set numerical and simulation results also reveal the great advantage of our proposed solutions for various types of ins in particular the proposed solutions are shown to outperform the algorithm developed by dahrouj and yu for the simpler case of the miso in

symmetry in a constrained hamiltonian system with singular higher order lagrangian for a canonical formalism with a higher order derivative the corresponding generalized first noether theorem gfnt for a constrained hamiltonian system and the generalized noether identities gni for a system with a non invariant action integral are derived which may be useful to analyse the dirac constraint for such a system using the gfnt another example is given in which dirac s conjecture fails using the gni the strong and weak conservation laws are deduced and it is pointed out that for certain variant systems there is also a dirac constraint suppose that there are only first class constraints fcc in a system then an algorithm for the construction of a gauge generator is developed once the hamiltonian and the fcc of the system with a higher order lagrangian are given

a numerical scheme for the pore scale simulation of crystal dissolution and precipitation in porous media in this paper we analyze a numerical scheme for a dissolution and precipitation model in porous media we focus here on the chemistry which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary we use a regularization approach for constructing a semi implicit scheme that is stable and convergent for dealing with the emerging time discrete nonlinear problems we propose a simple fixed point iterative procedure the paper is concluded by numerical results

stepsize control for delay differential equations using continuously imbedded runge kutta methods of sarafyan the use of continuously imbedded runge kutta sarafyan methods for the solution of ordinary differential equations with either time dependent or state dependent delays is discussed it is shown how to get reliable solutions for such problems in a manner that does not require that the effect of the local approximation error be considered separately from the local integration error it is also shown how to reliably handle derivative discontinuities that arise in the solution of differential equations with delays

an analysis of operator splitting techniques in the stiff case operator splitting methods are commonly used in many applications we focus here on the case where the evolution equations to be simulated are stiff we will more particularly consider the case of two operators a stiff one and a nonstiff one this occurs in numerous application fields e g combustion air pollution and reactive flows the classical analysis of the splitting error may then fail since the chosen splitting timestep t is in practice much larger than the fastest time scales the asymptotic expansion t 0 is therefore no longer valid we show here that singular perturbation theory provides an interesting framework for the study of splitting error some new results concerning the order of local errors are derived the main result deals with the choice of the sequential order for the operators the stiff operator must always be last in the splitting scheme

interpolation for variable order runge kutta methods abstract runge kutta methods approximate the solution of an initial value problem for a system of ordinary differential equations only on a mesh recent work considers how to approximate the solution everywhere variable order runge kutta methods are developed that have this capability one is about as efficient as a popular variable order method without this capability

operator splitting for nonlinear reaction diffusion systems with an entropic structure singular perturbation and order reduction in this paper we perform the numerical analysis of operator splitting techniques for nonlinear reaction diffusion systems with an entropic structure in the presence of fast scales in the reaction term we consider both linear diagonal and quasi linear non diagonal diffusion the entropic structure implies the well posedness and stability of the system as well as a tikhonov normal form for the nonlinear reaction term 23 it allows to perform a singular perturbation analysis and to obtain a reduced and well posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution we then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step t thus leading to the failure of the usual splitting analysis techniques we define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales all the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level we finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

addendum to a dynamical formalism of singular lagrangian system with higher derivatives a formulation for singular lagrangians with higher derivatives in a previous paper is supplemented and a complete formulation of canonical theory for the systems is presented it is shown that our previous formulation reproduces all constraints included in the ostrogradski transformations by using stationarity conditions of primary constraints secondarily on the effect of adding a total time derivative term to a nonsingular lagrangian with higher derivatives the previous paper is revised so as to give a correct formulation for counting the numbers of first and second class constraints it is concluded that the main part of our previous formulation needs no essential change

canonical formalism of action at a distance electrodynamics and many particle potential among charged particles the second post coulombian lagrangian of wheeler feynman electrodynamics for a many particle system is treated according to a canonical formalism of a singular lagrangian with higher derivatives the canonical equations are given in terms of a reduced hamiltonian with dirac brackets but they are transformed to be expressed in terms of ordinary poisson brackets by redefinition of canonical variables the reduced hamiltonian includes a characteristic form of three particle and four particle potentials finally a direct pathway to the reduced hamiltonian is presented via first order formalism of the maxwell theory with charged particles

higher order lagrangian systems geometric structures dynamics and constraints in order to study the connections between lagrangian and hamiltonian formalisms constructed from a perhaps singular higher order lagrangian some geometric structures are constructed intermediate spaces between those of lagrangian and hamiltonian formalisms partial ostrogradski s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined and some of their properties studied equations of motion constraints and arbitrary functions of lagrangian and hamiltonian formalisms are thoroughly studied in particular all the lagrangian constraints are obtained from the hamiltonian ones once the gauge transformations are taken into account the true number of degrees of freedom is obtained both in the lagrangian and hamiltonian formalisms and also in all the intermediate formalisms herein defined

stability properties of interpolants for runge kutta methods in solving initial value problems for ordinary differential equations by means of a discrete method we often need to know the solution on a set of points that differs from the grid systems with driving equation dense output discontinuity problems delay equations etc the nonnodal approximations are generally obtained by local interpolation and different features can be requested from the interpolants according to the problem treated in this paper we introduce the concept of stable interpolant for runge kutta methods roughly speaking a stable interpolant maintains the stability properties of the runge kutta method itself the lack of such stability makes the continuous extension not reliable especially in solving stiff equations

solution of structured non stiff odes abstract the solution of the initial value problem for a system of ordinary differential equations odes y f x y by runge kutta methods is considered the structure to be exploited is that a large portion of the cost of evaluating f x y arises in the evaluation of functions of the independent variable x only it is shown that when the structure is favorable there are methods which are considerably more efficient than those currently used in general purpose codes

convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods we have recently developed a generic approach to solving retarded and neutral delay differential equations ddes the approach is based on the use of an explicit continuous runge kutta formula and employs defect control the approach can be applied to problems with state dependent delays and vanishing delays in this paper we determine convergence properties for the numerical solution associated with methods that implement this approach we first analyze such properties for retarded ddes and then the analysis is extended to the case of neutral ddes ndes such an extension is particularly important since ndes have received little attention in the literature the main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect our analysis can also be applied directly to other numerical methods which use defect error control based on runge kutta formulas for ddes

redefinition of position variables and the reduction of higher order lagrangians single time lagrangians are treated in this paper describing the dynamics of systems of point particles which are given as formal power series in some ordering parameter and which may contain higher time derivatives in all terms but the leading one an efficient method for eliminating the higher time derivatives directly on the lagrangian level is presented this method clarifies the meaning of using the lower order equations of motion in higher order terms in a lagrangian the method consists of an iterative use of contact transformations in the jet prolongation of the extended configurations space and is called the method of redefinition of position variables several examples from electrodynamics and relativistic gravity are treated explicitly

runge kutta triples abstract the addition of a third runge kutta rk formula to an embedded pair forming an rk triple can yield an algorithm which gives solutions of appropriate asymptotic accuracy at points within the normal step intervals in the numerical solution of the initial value problem two modes of implementation are possible and it is shown that these can be regarded as equivalent when certain conditions are satisfied an optimized rk5 4 triple based on a particularly efficient embedded pair is presented and tested the results indicate the efficiency of the dense output technique