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

a derivative free variant called dfsa of dekkers and aarts continuous simulated annealing algorithm abstract we propose a derivative free implementation of dekkers and aarts continuous simulated annealing sa essential in dfsa is the random direction local technique that we introduce we study the convergence properties of the resulting algorithm and test its performance on a set of 50 problems numerical results are presented which show the robustness of the algorithm numerical comparisons with sa two derivative free simulated annealing algorithms and three population based global optimization algorithms show that the derivative free sa dfsa offers a reasonable alternative to some recent global optimization algorithms especially for problems requiring a direct search type algorithm

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

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

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

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

backward differentiation type formulas for volterra integral equations of the second kind numerical integration formulas are discussed which are obtained by differentiation of the volterra integral equation and by applying backward differentiation formulas to the resulting integro differential equation in particular the stability of the method is investigated for a class of convolution kernels the accuracy and stability behaviour of the method proposed in this paper is compared with that of i a block implicit runge kutta scheme and ii the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas the present method is particularly advantageous in the case of stiff volterra integral equations

differential evolution with generalized differentials in this paper we study the mutation operation of the differential evolution de algorithm in particular we propose the differential of scaled vectors called the generalized differential as opposed to the existing scaled differential vector in the mutation of de we derive the probability distribution of points generated by the mutation with generalized differentials we incorporate a vector projection based exploratory method within the new mutation scheme the vector projection is not mandatory and it is only invoked if trial points continue to be unsuccessful an algorithm is then proposed which implements the mutation strategy based on the difference of the scaled vectors as well as the vector projection technique a numerical study is carried out using a set of 50 test problems many of which are inspired by practical applications numerical results suggest that the new algorithm is superior to de

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

general instability results for interconnected systems general results giving conditions for an interconnected system to be input output and or lyapunov unstable are considered these results are derived in terms of the theory of dissipative systems this enables a very simple formulation of the requirements for instability in particular the restrictions of linearity and unstable subsystems that appear in previous results are seen to be unnecessary consequently the relationship between instability and stability conditions is made clearer a wide variety of useful instability criteria can be easily obtained as special cases

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

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

convergence properties of the runge kutta chebyshev method the runge kutta chebyshev method is ans stage runge kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations method of lines the method possesses an extended real stability interval with a length proportional tos 2 the method can be applied withs arbitrarily large which is an attractive feature due to the proportionality of withs 2 the involved stability property here is internal stability internal stability has to do with the propagation of errors over the stages within one single integration step this internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space time grid refinement for a model class of linear problems we prove convergence under the sole condition that the necessary time step restriction for stability is satisfied these error bounds are valid for anys and independent of the stiffness of the problem numerical examples are given to illustrate the theoretical results

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

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

pouzet runge kutta chebyshev method for volterra integral equations of the second kind we construct a stabilized runge kutta method of pouzet type pouzet runge kutta chebyshev method prkc for nonlinear stiff volterra integral equations of the second kind the prkc method is an explicit second order scheme which possesses extended stability region due to the first kind chebyshev polynomial used to establish the stability function

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

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

a numerical study of some modified differential evolution algorithms abstract modifications in mutation and localization in acceptance rule are suggested to the differential evolution algorithm for global optimization numerical experiments indicate that the resulting algorithms are considerably better than the original differential evolution algorithm therefore they offer a reasonable alternative to many currently available stochastic algorithms especially for problems requiring direct search type methods numerical study is carried out using a set of 50 test problems many of which are inspired by practical applications

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

lyapunov function for power system with transfer conductances a method is given for deriving a lyapunov function for a two machine power system with nonzero transfer conductances the assumptions required are only mildly restrictive and are likely to be satisfied for practical values of machine parameters the method shows promise for extension to systems of more than two machines

adaptive memetic differential evolution with global and local neighborhood based mutation operators differential evolution de is one of the most popular optimization methods for real valued problems and a large number of its variants have been proposed so far however bringing together different ideas that already led to successful de versions is rare in the literature in the present paper a novel de algorithm is proposed in which three among the most efficient concepts already applied separately within de framework are gathered together namely 1 the adaptation of algorithm control parameters and probabilities of using different mutation strategies 2 the use of nelder mead algorithm as a local search method hybridized with de and 3 the splitting mutation into global and local models when local mutation model is based on the concept of neighborhood of individuals organized on a ring topology the performance of the novel algorithm called adaptive memetic de with global and local neighborhood based mutation operators is compared with 13 different de variants on a set of 25 popular problems which include rotated shifted and hybrid composition functions it is found that although none de algorithm outperforms all the others for the majority of problems on average the proposed approach perform better than all 13 de algorithms selected for comparison the proposed algorithm is another heuristic approach developed to solve optimization problems the question may arise whether proposing novel methods is useful as no free lunch theorems for optimization state that the expected performance of all possible heuristics on all possible problems is equal in the last section of the paper the limitations and implications of no free lunch theorems are discussed based on rich but unfortunately frequently neglected literature a very simple continuous and differentiable minimization problem is proposed for which it is empirically verified that each among considered 14 de algorithms perform poorer than random sampling it is also empirically shown that when all considered de algorithms search for the maximum of the proposed problem they found lower minimum than de algorithms searching for the minimum such result is not unexpected according to no free lunch theorems and should be considered as a precaution from generalization of good performance of heuristic optimization methods

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

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

rkc time stepping for advection diffusion reaction problems the original explicit runge kutta chebyshev rkc method is a stabilized second order integration method for pure diffusion problems recently it has been extended in an implicit explicit manner to also incorporate highly stiff reaction terms this implicit explicit rkc method thus treats diffusion terms explicitly and the highly stiff reaction terms implicitly the current paper deals with the incorporation of advection terms for the explicit method thus aiming at the implicit explicit rkc integration of advection diffusion reaction equations in a manner that advection and diffusion terms are treated simultaneously and explicitly and the highly stiff reaction terms implicitly

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

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

scalar lyapunov function method for single and multimachine problems in power systems abstract a systematic method is proposed to derive appropriate lyapunov functions for single and multimachine problems in power systems that will enable suitable stability region estimates to be determined initially the lyapunov function for a simplified second order system is derived this function is then modified to form a suitable lyapunov function for the actual non linear system under study six examples for single and multimachine problems are considered

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