the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

runge kutta methods for the multi pantograph delay equation this paper deals with the properties of the numeric solution of the multi pantograph equationu t email protected t email protected 1u q 1t email protected 2u q 2t email protected lu q lt u 0 u 0 where 0

a highly stable deferred correction scheme with interpolant for systems of nonlinear two point boundary value problems iterated deferred correction is a very popular approach to the numerical solution of general first order systems of nonlinear two point boundary value problems typically the order of accuracy of the solution is increased by 2 each time that a deferred correction is applied and this allows inexpensive local error estimates to be computed using embedding recent research has shown how deferred correction schemes having the very desirable property of superconvergence can be derived this marks a significant step forward in the design of deferred correction algorithms although the price to be paid for superconvergence is that it is difficult to derive realistic embedded local error estimates in the present paper we show that richardson extrapolation is a viable alternative for error estimation and we consider in some detail the relative merits of embedding and extrapolation guided by this analysis we are able to derive for the first time a lobatto deferred correction code which is very efficient for the solution of stiff problems particularly in cases where the mirk code twpbvp f is unstable furthermore we are able to derive efficient interpolants for this lobatto code which are also applicable to mirk formulae and we consider the problem of estimating the error in these interpolants this completely solves the interpolation problem for our deferred correction codes apart perhaps for deriving interpolants for extremely stiff problems this new lobatto code is available on the web page of one of the present authors and fills a large gap in currently available deferred correction software

stability analysis for linear discretisations of the advection equation with runge kutta time integration for the 1 dim linear advection problem stability limits of runge kutta rk methods from 1st to 7th order in combination with upwind or centered difference schemes from 1st to 6th order are presented the analysis can be carried out in a rather general way by introduction of a broad class of runge kutta methods here called linear case runge kutta lc rk methods which behave completely similar for linear time independent and homogeneous ode systems and contain the classical order stage rk methods the set of conditions for the coefficients of these lc rk schemes could be derived explicitly for arbitrary order n from an efficiency viewpoint the lc rk 3rd order methods in combination with upwind 3rd or 5th order or the lc rk 4th order scheme with 4th order centered difference advection are a good choice the analysis can be extended easily to multidimensional splited advection for which a necessary stability condition is presented

attractors for the 3d baroclinic quasi geostrophic equations of large scale atmosphere abstract the existence of absorbing sets and a compact connected global attractor to the 3d baroclinic quasi geostrophic equations of large scale atmosphere are obtained using cft theory an upper bound of the hausdorff and fractal dimensions of the global attractor in terms of the physically relevant parameters re the reynolds number ro the rossby number and f is derived moreover similar results are also established for the 2d vorticity equations of globe atmosphere

a sufficient condition for gpn stability for delay differential equations in this paper the author considers a linear test delay differential equation with non constant coefficients related to the definitions of pn and gpn stability for numerical methods he defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for gpn stability

pullback attractors for closed cocycles abstract in this article using a result of pata and zelik 2007 45 we derive a general result on the existence of pullback attractors for closed cocycles acting on a banach space where the strong continuity is replaced by a much weaker requirement that the cocycle be a closed map as application we prove the existence of the pullback attractor of a cocycle associated with the z weak solutions of a non autonomous two dimensional primitive equations of the ocean

time point relaxation runge kutta methods for ordinary differential equations bellen a z jackiewicz and m zennaro time point relaxation runge kutta methods for ordinary differential equations journal of computational and applied mathematics 45 1993 121 137 we investigate convergence order and stability properties of time point relaxation runge kutta methods for systems of ordinary differential equations these methods can be implemented in gauss jacobi or gaussseidel modes denoted by trgjrk k or trgsrk k respectively where k stands for the number of picard lindelof iterations as k f 0 these modes tend to the same one step method for ordinary differential equations called diagonal split runge kutta dsrk method it is proved that these methods are convergent with order min k p where p is the order of the underlying runge kutta method recurrence relations resulting from application of trgjrk k trgsrk k and dsrk methods to the two dimensional test system u au pu u pu au t 3 0 where a and p are real parameters are derived and stability regions in the h p plane are plotted for some methods using a variant of the boundary locus method in most cases stability regions increase as the number of picard lindeldf iterations k becomes larger

a new collocation method for approximate solution of the pantograph functional differential equations with proportional delay the paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders the method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional differential equations with proportional delays including equations of neutral type the method is applicable to both initial and boundary value problems

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

a simple global model for the general circulation of the atmosphere in this article we present a new method of rossby asymptotics for the equations of the atmosphere similar to the geostrophic asymptotics we depart from the classical geostrophics see j g charney 5 and our previous article 29 by considering an asymptotics valid for the whole atmosphere not only in midlatitude regions and by taking into account the spherical form of the earth we obtain in this way a very simple global circulation model of the atmosphere for which the equations of motion for wind and temperature are linear evolution equations similar to the linear stokes equations furthermore the solutions are independent of longitude and winds travel exactly to the east or to the west in this mathematically oriented article we do not discuss the physical significance of the model that we derive except for observing that this picture coincides in general terms with the classically averaged data obtained by experimental measurements we also note that different global geostrophic asymptotics called planetary geostrophic asymptotics are considered elsewhere in the literature for example in j pedlosky 33 in a less mathematically rigorous way h lamb 21 observed that the only globally valid geostrophic flow that maintains a slow time scale is zonally symmetric see the comments in n phillips 35 and an explicit derivation in h jeffreys 19 c 1997 john wiley sons inc

a global code invariant under the higman sims group in this paper we consider certain doubly transitive permutation groups which have monomial representations which are reducible these groups have been studied by hale and shult 14 and by taylor i7 1 the stabilizer h of a vertex has a subgroup l of index two for which elements of l are not conjugate in g to elements of h l the monomial representations have two invariant subspaces which can be reduced modulo any prime to form codes over finite fields the invariant subspaces form global codes this approach is motivated by the work of ward 1 181 who considered the groups rslf2 q and sp 2n 4 the codes for psl 2 q give the classical extended quadratic residue codes in section 2 we construct the monomial representations and determine the two invariant subspaces taylor constructed 2 graphs from the centralizer algebra of these representations the invariant subspaces are spanned by vectors whose entries are parameters in these z graphs we illustrate the construction in the remaining sections with the doubiv transitive permutation representation on 176 vertices of the iiigman sims group this permutation representation was first discovered by graham higman the stabilizer of a vertex is a group h isomorphic to ptu 3 5 it

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

an approximate solution for a neutral functional differential equation with proportional delays in this paper a novel algorithm based on reproducing kernel theory for neutral functional differential equation with proportional delays is proposed the advantages of the presented method are the establishment of complete approximate solution theory and high precision of absolute error some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results

high accuracy large step explicit runge kutta hale rk schemes for computational aeroacoustics in many realistic calculations the computational grid spacing required to resolve the mean flow gradients is much smaller than the grid spacing required to resolve the unsteady propagating waves of interest because of this the high temporal resolution provided by existing optimized time marching schemes can be excessive due to the small time step required for stability in regions of clustered grid in this work explicit fourth order accurate runge kutta time marching schemes are optimized to increase the inviscid stability limit rather than the accuracy at large time steps single and multiple step optimized schemes are developed and analyzed the resulting schemes are validated on several realistic benchmark problems

comparison of monolithic and splitting solution schemes for dynamic porous media problems 5 summary proceeding from the governing equations describing a saturated poroelastic material with intrinsically 7 incompressible solid and fluid constituents we compare the monolithic and splitting solution of the different multi field formulations feasible in porous media dynamics because of the inherent solid fluid momentum 9 interactions one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint here the 11 resulting set of differential algebraic equations dae is solved by the finite element method fem following two different strategies 1 an implicit monolithic approach where the equations are first 13 discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators 2 a semi explicit implicit splitting scheme in the sense of a fractional step method 15 where the dae are first discretized in time split using intermediate variables and then discretized in space using linear equal order approximations for all primary unknowns finally a one and a two dimensional 17 wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies therefore several test cases differing in the used multi field formulation the 19 monolithic time stepping method and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter in the end we provide a 21 reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem copyright q 2009 john wiley sons ltd 23

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

a three stage vsvo hermite birkhoff taylor ode solver the new variable step variable order ode solver hbt p of order p presented in this paper combines a three stage runge kutta method of order 3 with a taylor series method of order p 2 to solve initial value problems y f t y y t 0 y 0 where y r r d and f rxr d r d the order conditions satisfied by hbt p are formulated and they lead to vandermonde type linear algebraic systems whose solutions are the coefficients in the formulae for hbt p a detailed formulation of variable step hbt p in both fixed order and variable order modes is presented the new method and the taylor series method have similar regions of absolute stability to obtain high accuracy results at high order this method has been implemented in multiple precision

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

parallel iteration across the steps of high order runge kutta methods for nonstiff initial value problems abstract for the parallel integration of nonstiff initial value problems ivps three main approaches can be distinguished approaches based on parallelism across the problem on parallelism across the method and on parallelism across the steps the first type of parallelism does not require special integration methods and can be exploited within any available ivp solver the method parallelism approach received much attention particularly within the class of explicit runge kutta methods originating from fixed point iteration of implicit runge kutta methods of gaussian type the construction and implementation on a parallel machine of such methods is extremely simple since the computational work per processor is modest with respect to the number of data to be exchanged between the various processors this type of parallelism is most suitable for shared memory systems the required number of processors is roughly half the order of the generating runge kutta method and the speed up with respect to a good sequential ivp solver is about a factor 2 the third type of parallelism step parallelism can be achieved in any ivp solver based on predictor corrector iteration and requires the processors to communicate after each full iteration if the iterations have sufficient computational volume then the step parallel approach may be suitable for implementation on distributed memory systems most step parallel methods proposed so far employ a large number of processors but lack the property of robustness due to a poor convergence behaviour in the iteration process hence the effective speed up is rather poor the dynamic step parallel iteration process proposed in the present paper is less massively parallel but turns out to be sufficiently robust to achieve speed up factors up to 15

order conditions for integrators and mappings with applications abstract we regard the elementary weight function as a mapping from input to output so that the mappings allow us to derive the order conditions for a given method more easily and concisely this paper presents the order conditions of one step runge kutta methods runge kutta type methods with derivatives a class of two step runge kutta methods and three step runge kutta methods based on the elementary weight mappings the composition formula of two mappings as well as the mappings i d r e defined on the set t of all rooted trees from an input to an output it is pointed out that the order conditions can be easily achieved for linear multistep methods and second derivative linear multistep methods of order p and it suffices to consider only bushy trees of order up to p together with the empty tree 0 for a linear multistep method or a second derivative linear multistep method of order p

on strong solutions of the multi layer quasi geostrophic equations of the ocean abstract in this article we study the multi layer quasi geostrophic equations of the ocean the existence of strong solutions is proved we also prove the existence of a maximal attractor in l 2 and we derive estimates of its hausdorff and fractal dimensions in terms of the data our estimates rely on a new formulation that we introduce for the multi layer quasi geostrophic equation of the ocean which replaces the nonhomogeneous boundary conditions and the nonlocal constraint on the stream function by a simple homogeneous dirichlet boundary condition this work improves the results given in c bernier existence of attractor for the quasi geostrophic approximation of the navier stokes equations and estimate of its dimension adv math sci appl 4 2 1994 465 489

stability analysis of runge kutta methods for unbounded retarded differential equations with piecewise continuous arguments abstract this paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments epca u t f t u t u t n t 0 u 0 u 0 where n z a sufficient condition that the differential equations are asymptotically stable is derived this paper is also concerned with the stability analysis of the runge kutta methods for equation u t au t bu t n t 0 u 0 u 0 where n z the conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given

numeric solutions for the pantograph type delay differential equation using first boubaker polynomials a numerical method is applied to solve the pantograph equation with proportional delay under the mixed conditions the method is based on the truncated first boubaker series the solution is obtained in terms of first boubaker polynomials also illustrative examples are included to demonstrate the validity and applicability of the technique the results obtained are compared by the known results

the hermitian two graph and its code abstract the two graph code of the hermitian two graph h q is investigated by use of this code we show that if q 3 mod 8 there exists no strongly regular graph with valency q 2 q 1 2 in the switching class corresponding to h q for q 5 the code is worked out in detail we generate the weight enumerator and give an account for the code words of several weights here it follows that there is a unique regular graph with valency 50 in the switching class

variational iteration method for solving a generalized pantograph equation the variational iteration method is applied to solve the generalized pantograph equation this technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of lagrange multipliers for identification of optimal value of a parameter in a functional employing this technique it is possible to find the exact solution or an approximate solution of the problem some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results

binary codes derived from the hoffman singleton and higman sims graphs some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the hoffman singleton graph and the higman sims graph some of the codes are optimal or nearly optimal for the given length and dimension the dual codes admit majority logic decoding

stability of generalized runge kutta methods for stiff kinetics coupled differential equations a stability and efficiency improved class of generalized runge kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and or nonlinear coupled differential equations the determination of the coefficients required by the method is precisely obtained from the so called equations of condition which in turn are derived by an approach based on butcher series since the equations of condition are fewer in number free parameters can be chosen for optimizing any desired feature of the process a further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced in addition the a stability properties of the method are investigated implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

estimating the error of the classic runge kutta formula abstract a high quality error estimator is presented for the classic four stage fourth order runge kutta formula that is free when using this estimator the solution can be approximated accurately throughout the step at no additional cost

cluster expansion for abstract polymer models new bounds from an old approach we revisit the classical approach to cluster expansions based on tree graphs and establish a new convergence condition that improves those by koteck preiss and dobrushin as we show in some examples the two ingredients of our approach are i a careful consideration of the penrose identity for truncated functions and ii the use of iterated transformations to bound tree graph expansions

stochastic approach for the solution of multi pantograph differential equation arising in cell growth model in this paper a computational technique is introduced for the solution of the first order multi pantograph differential equation mpde through some well known optimization algorithms like sequential quadratic programming sqp and active set technique ast furthermore artificial neural network ann is used for networking of the first order multi pantograph differential equation in used to provide mathematical model based on unsupervised error for equation moreover mathematical modeling has been performed perfectly through multi runs for simulation to justify the better convergence of the solutions also two examples are presented to exhibit the aptitude of the method sqp and ast the comparative study will be made with reported techniques such as variational iteration technique vit 6 and collocation based on bernstein polynomial method bcm 6

a one step 7 stage hermite birkhoff taylor ode solver of order 11 a one step 7 stage hermite birkhoff taylor method of order 11 denoted by hbt 11 7 is constructed for solving nonstiff first order initial value problems y f t y y t 0 y 0 the method adds the derivatives y to y 6 used in taylor methods to a 7 stage runge kutta method of order 6 forcing an expansion of the numerical solution to agree with a taylor expansion of the true solution to order 11 leads to taylor and runge kutta type order conditions these conditions are reorganized into vandermonde type linear systems whose solutions are the coefficients of the method the new method has a larger scaled interval of absolute stability than the dormand prince dp87 and a larger unscaled interval of absolute stability than the taylor method t11 of order 11 hbt 11 7 is superior to dp87 and t11 in solving several problems often used to test higher order ode solvers on the basis of the number of steps cpu time and maximum global error numerical results show the benefit of adding high order derivatives to runge kutta methods

optimal control of the primitive equations of the ocean with state constraints abstract we investigate in this article pontryagin s maximum principle for a class of control problems associated with the primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in wang 2002 7 for the three dimensional navier stokes ns equations the main difference between this work and wang 2002 7 is that the nonlinearity in the pes is stronger than in the three dimensional ns systems

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

parallel iterated runge kutta methods for stiff ordinary differential equations abstract for the numerical integration of a stiff ordinary differential equation fully implicit runge kutta methods offer nice properties like a high classical order and high stage order as well as an excellent stability behaviour however such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task this paper discusses an iteration scheme to tackle this problem by means of a suitable choice of the iteration parameters the implicit relations for the stage values as they occur in each iteration can be uncoupled so that they can be solved in parallel the resulting scheme can be cast into the class of diagonally implicit runge kutta dirk methods and similar to these methods requires only one lu factorization per step per processor the stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed we discuss several choices to obtain good stability and fast convergence based on these approaches we wrote two codes possessing local error control and stepsize variation we have implemented both codes on an alliant fx 4 machine four parallel vector processors and shared memory and measured their speedup factors for a number of test problems furthermore the performance of these codes is compared with the performance of the best stiff ode codes for sequential computers like simple lsode and radau5

a two grid finite difference method for the primitive equations of the ocean abstract a two level method is proposed for the finite difference approximation of a class of dissipative systems including the primitive equations of the ocean the two level or multilevel method amounts to solving a nonlinear problem on the single coarsest grid subsequent approximations are generated on a succession of refined meshes by solving a linearized problem about the solution on the previous level the stability analysis for an euler type scheme shows that the multilevel method can allow larger time steps than the traditional one level methods some numerical simulations of the primitive equations of the ocean using this new multi scale algorithm are presented to illustrate the method

partitioned krylov subspace iteration in implicit runge kutta methods abstract application of the method of lines to partial differential equation leads to very large sparse systems of ordinary differential equations which are usually stiff we consider high order implicit runge kutta methods for the time integration of these systems and propose a partitioned krylov method for the solution of the resulting linear systems it is shown that this method allows for parallel computation in the construction of krylov subspaces and in a few examples it converges much faster than gmres

stability of the discretized pantograph differential equation in this paper we study discretizations of the general pantograph equation y t ay t by 6 t cy o t t 0 y o yo where a b c and yo are complex numbers and where 0 and 0 are strictly increasing functions on the nonnegative reals with 6 0 q o 0 and 0 t t 0 t t for positive t our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations and we will identify conditions on a b c and the stepsize which imply that the solution sequence yn i is bounded or that it tends to zero algebraically as a negative power of n

pontryagin s maximum principle for optimal control of the stationary primitive equations of the ocean abstract we study in this article pontryagin s maximum principle for a class of control problems associated with the stationary primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in g wang pontryagin s maximum principle for optimal control of the stationary navier stokes equations nonlinear anal 52 2003 1853 1866 in the case of the three dimensional navier stokes ns equations the main difference between this work and the one cited above is that the nonlinearity in pes is stronger than that in the three dimensional ns equations

e approximate polynomial solutions for the multi pantograph equation with variable coefficients abstract in this paper we present a new method to obtain analytical approximate polynomial solutions for the multi pantograph equation this method allows us to determine an approximate polynomial solution of the nonlinear multi pantograph equation with variable coefficients and to specify the maximal error of the approximation relative to the exact solution of the problem two numerical examples are given to illustrate the validity and the applicability of the method a comparison with previous results is also presented

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

a general strategy for the optimization of runge kutta schemes for wave propagation phenomena we analyze optimized explicit runge kutta schemes rk for computational aeroacoustics and wave propagation phenomena in general exploiting the analysis developed in s pirozzoli performance analysis and optimization of finite difference schemes for wave propagation problems j comput phys 222 2007 809 831 we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics and provide a general strategy to design runge kutta methods tailored for specific applications we present families of optimized second and third order runge kutta schemes with up to seven stages and describe their implementation in the framework of williamson s 2n storage formulation j h williamson low storage runge kutta schemes j comput phys 35 1980 48 56 numerical simulations of the 1d linear advection equation and of the 2d linearized euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes

averaging of the planetary 3d geostrophic equations with oscillating external forces abstract in this article we consider a non autonomous three dimensional planetary geostrophic model of the ocean with a singularly oscillating external force depending on a small parameter we prove the existence of the uniform global attractor a furthermore using the method of 11 in the case of the two dimensional navier stokes systems we study the convergence of a as goes to zero

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

explicit methods based on a class of four stage fourth order runge kutta methods for preserving quadratic laws this paper concerns the study of explicit 4 stage fourth order runge kutta methods which preserve quadratic conservation laws when their weights are made solution dependent application to orthogonal and hamiltonian linear systems and numerical tests are also reported

discretizations of nonlinear differential equations using explicit finite order methods abstract in this work we study the appearance of spurious solutions when first order differential equations with unimodal right hand sides are discretized using runge kutta schemes these spurious solutions are explained in terms of the iteration functions schemes that produce good approximating solutions for much longer times are given

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

high order taylor galerkin methods for linear hyperbolic systems a new family of high order taylor galerkin schemes is presented for the analysis of first order linear hyperbolic systems the schemes are unconditionally stable which makes them very attractive to use in conjunction with adaptive hp finite element methods for spatial approximation

linear codes and doubly transitive symmetric designs abstract an application of linear codes for the construction of a 2 transitive symmetric design from its residual or derived designs is discussed the higman design is constructed as the design supported by the minimum weight codewords in the extended binary code of certain designs invariant under p u 3 5 2