triangularly implicit iteration methods for ode ivp solvers it often happens that iteration processes used for solving the implicit relations arising in ode ivp methods only start to converge rapidly after a certain number of iterations fast convergence right from the beginning is particularly important if we want to use so called step parallel iteration in which the iteration method is concurrently applied at a number of step points in this paper we construct highly parallel iteration methods that do converge fast from the first iteration on our starting point is the pdirk method parallel diagonally implicit iterated runge kutta method designed for solving implicit runge kutta equations on parallel computers the pdirk method may be considered as a newton type iteration in which the newton jacobian is simplified to block diagonal form however when applied in a step parallel mode it turns out that its relatively slow convergence or even divergent behavior reduces the effectiveness of the step parallel scheme by replacing the block diagonal newton jacobian approximation in pdirk by a block triangular approximation we do achieve convergence right from the beginning at a modest increase of the computational costs our convergence analysis of the block triangular approach will be given for the wide class of general linear methods but the derivation of iteration schemes is limited to runge kutta based methods a number of experiments show that the new parallel triangularly implicit iterated runge kutta method ptirk method is a considerable improvement over the pdirk method

improving the efficiency of the iterative schemes for implicit runge kutta methods in this paper we consider a class of iterative schemes for implicit runge kutta methods substituting some of the iterations by another with a lower cost that does not require any evaluation of the derivative function we obtain a modified scheme study its properties of convergence and show the efficiency of the algorithm by some numerical experiments

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

an iterated radau method for time dependent pdes this paper is concerned with the time integration of semi discretized multi dimensional pdes of advection diffusion reaction type to cope with the stiffness of these odes an implicit method has been selected viz the two stage third order radau iia method the main topic of this paper is the efficient solution of the resulting implicit relations first a modified newton process has been transformed into an iteration process in which the 2 stages are decoupled and moreover can exploit the same lu factorization of the iteration matrix next we apply a so called approximate matrix factorization amf technique to solve the linear systems in each newton iteration this amf approach is very efficient since it reduces the multi dimensional system to a series of one dimensional systems the total amount of linear algebra work involved is reduced enormously by this approach the idea of applying amf to two dimensional problems is quite old and goes back to peaceman and rachford in the early fifties the situation in three space dimensions is less favourable and will be analyzed here in more detail both theoretically and experimentally furthermore we analyze a variant in which the amf technique has been used to really solve until convergence the underlying radau iia method so that we can rely on its excellent stability and accuracy characteristics finally the method has been tested on several examples also a comparison has been made with the existing codes vodpk and imexrkc and the efficiency cpu time versus accuracy is shown to be at least competitive with the efficiency of these solvers

iterative schemes for gauss methods abstract in this paper we consider a class of iterative schemes for implicit runge kutta methods and we study the convergence of these schemes for a family of nonlinear stiff problems a particular convergent scheme for the two stage gauss method is proposed and the order and linear stability properties are analyzed finally some numerical experiments are included in order to show the efficiency of the method

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

on the implementation of singly implicit runge kutta methods a modified newton method is often used to solve the algebraic equations that arise in the application of implicit runge kutta methods when the runge kutta method has a coefficient matrix a with a single point spectrum with eigenvalue a the efficiency of the modified newton method is much improved by using a similarity transformation of a each iteration involves vector transformations in this article an alternative iteration scheme is obtained which does not require vector transformations and which is simpler in other respects also both schemes converge in a finite number of iterations when applied to linear systems of differential equations but the new scheme uses the nilpotency of a ai to achieve this numerical results confirm the predicted convergence for nonlinear systems and indicate that the scheme may be a useful alternative to the modified newton method for low dimensional systems the scheme seems to become less effective as the dimension increases however it has clear advantages for parallel computation making it competitive for high dimensional systems

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

algebraically stable and implementable runge kutta methods of high order there are three interesting properties of methods for stiff ordinary differential equations order stability and efficiency of implementation this paper constructs runge kutta methods of orders 5 and 6 which possess these properties to a high extent we further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods

an efficient method for solving implicit and explicit stiff differential equations when the s stage fully implicit runge kutta rk method is used to solve a system of n ordinary differential equations ode the resulting algebraic system has a dimension ns its solution by gauss elimination is expensive and requires 2s3n3 3 operations in this paper we present an efficient algorithm which differs from the traditional rk method the formal procedure for uncoupling the algebraic system into a block diagonal matrix with s blocks of size n is derived for any s its solution is s2 2 times faster than the original nondiagonalized system for s even and s3 s 1 for s odd in terms of number of multiplications as well as s2 times in terms of number of additions multiplications in particular for s 3 the method has the same precision and stability properties as the well known rk based radauiia quadrature of ehle implemented by hairer and wanner in radau5 algorithm unlike radau5 however the method is applicable with any s and not only to the explicit odes my f x y where m const but also to the general implicit odes of the form f x y y 0 the block diagonal form of the algebraic system allows parallel processing the algorithm formally differs from the implicit rk methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coefficients are found by enforcing y to satisfy the differential equation at the collocation points locations of those points are found from the derived stability function such as to guarantee either a or l stability properties as well as a superior precision of the algorithm if constructed such as to be l stable the method is a good candidate for solving differential algebraic equations daes although not limited to any specific field the application of the method is illustrated by its implementation in the multibody dynamics described by both odes and daes copyright 2000 john wiley sons ltd

optimal resilient threshold gq signatures gq signature scheme is widely used in many cryptographic protocols such as forward secure signature scheme identity based signature scheme etc however there is no threshold version of this important signature scheme in the open literature we proposed the first threshold gq signature scheme the scheme is proved unforgeable and robust against any adaptive adversary by assuming hardness of computing discrete logarithm modulo a safe prime and existence of fully synchronous broadcast channel furthermore with some modifications our scheme achieves optimal resilience such that the adversary can corrupt up to a half of the players as an extension of our work we provided a threshold identity based signature scheme and a threshold forward secure signature scheme which is the threshold version of the most efficient forward secure signature scheme up to now

analysis of energy and quadratic invariant preserving equip methods abstract in this paper we are concerned with the analysis of a class of geometric integrators at first devised in brugnano et al 2010 and brugnano et al 2012 which can be regarded as an energy conserving variant of gauss collocation methods with these latter they share the property of conserving quadratic first integrals but in addition they also conserve the hamiltonian function itself we here reformulate the methods in a more convenient way and propose a more refined analysis than that given in brugnano et al 2012 also providing as a by product a practical procedure for their implementation a thorough comparison with the original gauss methods is carried out by means of a few numerical tests solving hamiltonian and poisson problems

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

blended implicit methods for the numerical solution of dae problems recently a new approach for solving the discrete problems generated by the application of block implicit methods for the numerical solution of initial value problems for odes has been devised l brugnano blended block bvms b 3vms a family of economical implicit methods for odes j comput appl math 116 2000 41 62 l brugnano c magherini blended implementation of block implicit methods for odes appl numer math 42 2002 29 45 l brugnano d trigiante block implicit methods for odes in d trigiante ed recent trends in numerical analysis nova science publishers new york 2001 pp 81 105 this approach is based on the so called blended implementation of the methods giving corresponding blended implicit methods the latter have been implemented in the computational code bim l brugnano c magherini the bim code for the numerical solution of odes j comput appl math 164 165 2004 145 158 blended implicit methods are here extended to handle the numerical solution of dae problems resulting in a straightforward generalization of the basic approach

a note on efficient preconditioner of implicit runge kutta methods with application to fractional diffusion equations abstract this paper is concerned with the construction of efficient preconditioning method for systems arising from implicit runge kutta discretizations of time dependent pdes a polynomial preconditioner is proposed based on the kronecker product splitting iteration method introduced by chen bit 54 607 721 2014 some useful properties on the spectrum of the preconditioned matrix are established numerical experiments concerning discrete solution of fractional diffusion equations are presented to show the effectiveness of the proposed polynomial preconditioner

stage parallel fully implicit rungekutta solvers for discontinuous galerkin fluid simulations in this paper we develop new techniques for solving the large coupled linear systems that arise from fully implicit rungekutta methods this method makes use of the iterative preconditioned gmres algorithm for solving the linear systems which has seen success for fluid flow problems and discontinuous galerkin discretizations by transforming the resulting linear system of equations one can obtain a method which is much less computationally expensive than the untransformed formulation and which compares competitively with other time integration schemes such as diagonally implicit rungekutta dirk methods we develop and test several ilu based preconditioners effective for these large systems we additionally employ a parallel in time strategy to compute the rungekutta stages simultaneously numerical experiments are performed on the navierstokes equations using euler vortex and 2d and 3d naca airfoil test cases in serial and in parallel settings the fully implicit radau iia rungekutta methods compare favorably with equal order dirk methods in terms of accuracy number of gmres iterations number of matrixvector multiplications and wall clock time for a wide range of time steps

stability of collocation at gaussian points symmetric runge kutta schemes are particularly useful for solving stiff two point boundary value problems such a stable schemes perform well in many cases but it is demonstrated that in some instances the stronger property of algebraic stability is required a characterization of symmetric algebraically stable runge kutta schemes is given the class of schemes thus defined turns out not to be very rich the only collocation schemes in it are those based on gauss points and other schemes in the class do not seem to offer any advantage over collocation at gaussian points

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

dynamical properties of hybrid systems simulators this paper analyzes the dynamical properties of a class of hybrid systems simulators a hybrid system is a dynamical system with a state that can both flow and jump its simulator attempts to generate its solutions approximately the paper presents mild regularity conditions on the hybrid system and its simulator to guarantee that simulated solutions are close to actual solutions on compact hybrid time intervals and that asymptotically stable compact sets are preserved in a semiglobal practical sense under simulation in fact it is established that asymptotically stable compact sets are continuous in the integration step size parameter of the simulator that is as the step size of the simulator converges to zero the asymptotically stable set observed in simulations approaches the asymptotically stable compact set of the true hybrid system examples are used to illustrate concepts and results

second derivative methods applied to implicit first and second order systems a wide variety of methods have been proposed for solving implicit second order systems of differential equations such as arise in structural and mechanical vibration analyses here we discuss a class of methods based on second derivative formulae some of which have previously been used by enright 1 we show them to have desirable analytic and implementation properties we relate these methods to others recently proposed by brusa and nigro and by serbin showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors since implicit first order systems are also important in practice for example in finite element analyses of heat conduction problems we demonstrate the efficient implementation of the second derivative formulae in this case too

diffusion of a chemically reactive species of a power law fluid past a stretching surface abstract a numerical solution for the steady magnetohydrodynamic mhd non newtonian power law fluid flow over a continuously moving surface with species concentration and chemical reaction has been obtained the viscous flow is driven solely by the linearly stretching sheet and the reactive species emitted from this sheet undergoes an isothermal and homogeneous one stage reaction as it diffuses into the surrounding fluid using a similarity transformation the governing non linear partial differential equations are transformed into coupled nonlinear ordinary differential equations the governing equations of the mathematical model show that the flow and mass transfer characteristics depend on six parameters namely the power law index the magnetic parameter the local grashof number with respect to species diffusion the modified schmidt number the reaction rate parameter and the wall concentration parameter numerical solutions for these coupled equations are obtained by the keller box method and the solutions obtained are presented through graphs and tables the numerical results obtained reveal that the magnetic field significantly increases the magnitude of the skin friction but slightly reduces the mass transfer rate however the surface mass transfer strongly depends on the modified schmidt number and the reaction rate parameter it increases with increasing values of these parameters the results obtained reveal many interesting behaviors that warrant further study of the equations related to non newtonian fluid phenomena especially shear thinning phenomena shear thinning reduces the wall shear stress

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

order optimal preconditioners for implicit runge kutta schemes applied to parabolic pdes in this paper we show that standard preconditioners for parabolic pdes discretized by implicit euler or crank nicolson schemes can be reused for higher order fully implicit runge kutta time discretization schemes we prove that the suggested block diagonal preconditioners are order optimal for a stable and irreducible runge kutta schemes with invertible coefficient matrices the theoretical investigations are confirmed by numerical experiments

time consistent fluid structure interaction on collocated grids for incompressible flow abstract consistent time integration on collocated grids for incompressible flow has been studied for static grids using the piso method in which the dependencies on time step size and under relaxation has been studied in detail however for moving grids a detailed description is still missing therefore a step by step analysis of a time consistent fluid structure interaction fsi method for incompressible flow on collocated grids is presented the method consist of face normal and area correction for moving grids treatment of velocity boundary conditions for no slip walls time integration of structure equations and fluid force interpolation to structure the basis of the method is the piso method of the incompressible navier stokes equations time consistency on static grids is shown first after which time consistency on moving grids is described and analyzed for moving grids consistent time integration is described in two ways 1 constructing the face velocities from a normal and tangential component and 2 correcting the face flux with a face normal and face area correction for both descriptions the general formulation for the backward differencing schemes bdf are given and the correct behavior of the first second and third order schemes is demonstrated by means of an academic test case circular cavity flow additionally the force coupling from the fluid to the structure is discussed in detail for combining a fourth order explicit runge kutta scheme for the structure with a bdf scheme for the fluid three interpolations for the fluid forces are shown which result in either a first order or second order fsi scheme third order fsi is demonstrated when the third order bdf scheme is applied on both the structure and fluid equations also under relaxation for the fluid equations is considered and it is demonstrated that the order of the three bdf schemes are independent of the under relaxation factor finally the proposed method of time consistent fsi on collocated grids for incompressible flows is demonstrated by applying it to a three dimensional flow over an elastic structure in a channel and the cylinder flap fsi benchmark case of turek and hron

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

on diagonally iterated runge kutta methods for dissipative odes abstract in this paper we show that for most classical runge kutta methods there exist diagonally implicit schemes that are convergent for the class of dissipative problems for singly implicit methods it is shown that the iteration of van der houwen and sommeijer 1991 is also convergent in this context however numerical results indicate that the schemes obtained for the gauss and radau iia methods are far less efficient than those obtained in van der houwen and sommeijer 1991

a multisecret sharing scheme for color images based on cellular automata in this work a new multisecret sharing scheme for secret color images among a set of users is proposed the protocol allows that each participant in the scheme to share a secret color image with the rest of participants in such a way that all of them can recover all the secret color images only if the whole set of participants pools their shadows the proposed scheme is based on the use of bidimensional reversible cellular automata with memory the security of the scheme is studied and it is proved that the protocol is ideal and perfect and that it resists the most important statistical attacks

parallel ode solvers with stepsize control in this paper we propose a parallel implementation of one step methods with stepsize control for the numerical solution of ivps for odes of the form y t f t y t y t 0 y 0 the proposed implementation is based on the fact that any one step ode method on a mesh t 0 n y 0 known in a previous paper 1989 we introduced a paral iterative algorithm for the approximation of the trajectory y 0 y 1 y n in which a block of guessed values u 0 0 y 0 u 0 1 u 0 n is iterated concurrently with respect to the index n until an error proportional to a given iteration tolerance tol i t is reached here that parallel algorithm is developed further in order to perform the stepsize control strategy according to a given step tolerance tol s t moreover an analysis of the optimal ratio between tol i t and tol s t is given the paper ends with numerical examples and estimations of the attainable speedup

an efficient family of strongly a stable runge kutta collocation methods for stiff systems and daes part i stability and order results for each integer s 3 a new uniparametric family of stiffly accurate strongly a stable s stage runge kutta methods is obtained these are collocation methods with a first internal stage of explicit type the methods are based on interpolatory quadrature rules with precision degree equal to 2s 4 and all of them have two prefixed nodes c 1 0 and c s 1 the amount of implicitness of our s stage method is similar to that involved with the s stage lobattoiiia method or with the s 1 stage radauiia method the new family of runge kutta methods proves to be of interest for the numerical integration of stiff systems and differential algebraic equations in fact on several stiff test problems taken from the current literature two methods selected in our 4 stage family seem to be slightly more efficient than the 3 stage radauiia method and also more robust than the 4 stage lobattoiiia method

speeding up netwton type iterations for stiff problems iterative schemes based on the cooper and butcher iteration 5 are considered in order to implement highly implicit runge kutta methods on stiff problems by introducing two appropriate parameters in the scheme a new iteration making use of the last two iterates is proposed specific schemes of this type for the gauss radau ia iia and lobatto iiia b c processes are developed it is also shown that in many situations the new iteration presents a faster convergence than the original

robust threshold dss signatures we present threshold dss digital signature standard signatures where the power to sign is shared by n players such that for a given parameter t

new t n threshold directed signature scheme with provable security directed signature scheme allows only a designated verifier to check the validity of the signature issued to him and at the time of trouble or if necessary any third party can verify the signature with the help of the signer or the designated verifier as well due to its merits directed signature scheme is widely used in situations where the receiver s privacy should be protected threshold directed signature is an extension of the standard directed signature in which several signers may be required to cooperatively sign messages for sharing the responsibility and authority to the best of our knowledge threshold directed signature has not been well studied till now therefore in this paper we would like to formalize the threshold directed signature and its security model then present a new t n threshold directed signature scheme from bilinear pairings and use the techniques from provable security to analyze its security

a and b stability for runge kutta methods characterizations and equivalence using a special representation of runge kutta methods w transformation simple characterizations ofa stability andb stability have been obtained in 9 8 7 in this article we will make this representation and their conclusions more transparent by considering the exact runge kutta method finally we demonstrate by a numerical example that for difficult problemsb stable methods are superior to methods which are only a stable

on the boundary layer equations for power law fluids we reconsider the problem of the boundary layer flow of a non newtonian fluid whose constitutive law is given by the classical ostwald de waele power law model the boundary layer equations are solved in similarity form the resulting similarity solutions for shear thickening fluids are shown to have a finite width crisis resulting in the prediction of a finite width boundary layer a secondary viscous adjustment layer is required in order to smooth out the solution and to ensure correct matching with the far field boundary conditions in the case of shear thinning fluids the similarity forms have solutions whose decay into the far field is strongly algebraic smooth matching between these inner algebraically decaying solutions and an outer uniform flow is achieved via the introduction of a viscous diffusion layer

the role of orthogonal polynomials in numerical ordinary differential equations abstract orthogonal polynomials have many applications to numerical ordinary differential equations some of these especially those connected with the quadrature formulae on which many differential equation methods are based are to be expected on the other hand there are many surprising applications quite unlike traditional uses of orthogonal polynomials this paper surveys many of these applications especially those related to accuracy and implementability of runge kutta methods

the dichotomy of stiffness pragmatism versus theory the problem of stiffness occuring in ordinary differential equations is now well understood and various concepts such as nonlinear stability b convergence and stiff order have been introduced as being relevant to an understanding of the behaviour of numerical methods when applied to stiff problems however it is often the case that some of these properties are not always attainable when methods are implemented efficiently and in this paper we will investigate this paradox in some detail in addition we will give a resume of some of the recent theoretical developments in the applicability of numerical methods to stiff problems and make some comparisons between existing codes based on runge kutta and linear multistep methods we will conclude this discussion with the introduction of a very general family of methods called multivalue methods and show how the important concepts of cheap implementation stage order and stability apply to these methods

using krylov methods in the solution of large scale differential algebraic systems in this paper a new algorithm for the solution of large scale systems of differential algebraic equations is described it is based on the integration methods in the solver dassl but instead of a direct method for the associated linear systems which arise at each time step we apply the preconditioned gmres iteration in combination with an inexact newton method the algorithm along with those in dassl is implemented in a new solver called daspk we outline the algorithms and strategies used and discuss the use of the solver we develop and analyze some preconditioners for a certain class of dae stems and finally demonstrate the application of daspk on two example problems

on the numerical solution of stiff ivps by lobatto iiia runge kutta methods in this paper the numerical solution of stiff initial value problems by means of lobatto iiia runge kutta methods is considered new iterative schemes for an efficient solution of the implicit equations associated to the fourth and sixth order runge kutta lobatto iiia formulae are proposed finally the results of some numerical experiments are presented to show that when implemented with the new iterative schemes lobatto iiia formulae can be competitive with standard codes used in solving stiff systems

a note on magnetohydrodynamic flow of a power law fluid over a stretching sheet this paper presents a numerical study of the flow of an electrically conducting power law fluid in the presence of a uniform transverse magnetic field this flow is governed by the non linear differential equationn f n 1 f f 2 2nn 1ff mf 0where a prime denotes differentiation with respect to the similarity variable h and n m are respectively the power law index and the magnetic parameter in this work the numerical solutions are obtained using a runge kutta algorithm for high order initial value problems

constructive characterization ofa stable approximations to exp z and its connection with algebraically stable runge kutta methods all rational approximations to exp z of order 2m m denotes the maximal degree of nominator and denominator are given by a closed formula involving real parameters using the theory of order stars 9 necessary and sufficient conditions fora stability respectivelyi stability are given on the basis of this characterization relations between the concepts ofa stability and algebraic stability for implicit runge kutta methods are investigated in particular we can partly prove the conjecture that to any irreduciblea stabler z of oderp 0 there exist algebraically stable runge kutta methods of the same order withr z as stability function

gauss seidel iteration for stiff odes from chemical kinetics a simple gauss seidel technique is proposed that exploits the special form of the chemical kinetics equations classical aitken extrapolation is applied to accelerate convergence the technique is meant for implementation in stiff solvers that are used in long range transport air pollution codes using operator splitting splitting necessarily gives rise to a great deal of integration restarts because the gauss seidel iteration works matrix free it has much less overhead than the modified newton method start up costs therefore can be kept low with this technique preliminary promising numerical results are presented for a prototype of a second order backward differentiation formula bdf solver applied to a stiff ordinary differential equation ode from atmospheric chemistry a favourable comparison with the general purpose bdf code dassl is included the matrix free technique may also be of interest for other chemically reacting fluid flow problems

performance of gauss implicit runge kutta methods on separable hamiltonian systems abstract we consider implementations of a variable step size and separately constant step size fourth order symplectic gauss implict runge kutta method for the solution of hamiltonian systems we test our implementations on kepler s problem with the aim of judging the algorithms qualitative behavior and efficiency in particular we introduce compensated summation as a method of controlling roundoff accumulation also we show how the variable step size gauss implicit runge kutta method performs on kepler s problem with solution orbits of high eccentricity and compare its performance with that of two runge kutta nystrom codes finally we discuss the calculation of efficient starting values for the associated iterations measure the cost in iterations of our various predictors and comment on the strategies for terminating the iteration

on the stability of volterra runge kutta methods we investigate the stability of extended runge kutta methods for volterra integral equations of the second kind as test equations we choose linear convolution equations with positive definite l 1 kernels our results are strongly connected with the contractivity of runge kutta methods for nonlinear dissipative differential equations

construction of runge kutta type methods for solving ordinary differential equations in this paper we study runge kutta methods with continuous stage introduced by butcher in 1987 by setting the coefficients of this family of methods and choosing the appropriate numerical integration we derive new classes of runge kutta methods furthermore we extend the w transformation technique by permitting w to be a non square matrix this allows us to construct more high order implicit runge kutta methods with some geometric properties specially we provide runge kutta methods with continuous stage which are conjugate symplectic symmetric or energy preserving for solving hamiltonian systems we also present some numerical experiments to verify our results

a parallelizable preconditioner for the iterative solution of implicit runge kutta type methods the main diculty in the implementation of most standard implicit runge kutta irk methods applied to sti ordinary dierential equations odes is to eciently solve the nonlinear system of equations in this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit euler method the preconditioner is based on the w transformation of the rk coecient matrices discovered by hairer and wanner for sti odes the preconditioner is by construction asymptotically exact for methods with an invertible rk coecient matrix the methodology is particularly useful when applied to super partitioned additive runge kutta spark methods the nonlinear system can be solved by inexact simplied newton iterations at each simplied newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system c 1999 published by elsevier science b v all rights reserved

feasibility and contractivity in implicit runge kutta methods abstract this paper is concerned with implicit runge kutta methods for the numerical solution of initial value problems in ordinary differential equations for these methods a review is presented of the fundamental concept of algebraic stability introduced in 1979 independently by burrage butcher and by crouzeix furthermore a new framework is given in which algebraic stability becomes equivalent to both contractivity and feasibility of a runge kutta method

structure preservation for constrained dynamics with super partitioned additive runge kutta methods a broad class of partitioned differential equations with possible algebraic constraints is considered including hamiltonian and mechanical systems with holonomic constraints for mechanical systems a formulation eliminating the coriolis forces and closely related to the euler lagrange equations is presented a new class of integrators is defined the super partitioned additive runge kutta spark methods this class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms a linear stability and convergence analysis of these methods is given spark methods allowing the direct preservation of certain properties are characterized different structures and invariants are considered the manifold of constraints symplecticness reversibility contractivity dilatation energy momentum and quadratic invariants with respect to linear stability and structure preservation the class of s stage lobatto iiia b c c spark methods is of special interest controllable numerical damping can be introduced by the use of additional parameters some issues related to the implementation of a reversible variable stepsize strategy are discussed

efficient iterations for gauss methods on second order problems we consider some important aspects about the implementation of high order implicit formulas specially the gauss methods for solving second order differential systems having high frequencies and small amplitudes superimposed the choice of an appropriate iterative scheme is discussed in detail important topics about the predictors initial guesses are analyzed and a variable order strategy to select the best predictor at each integration step is supplied a few numerical experiments on some standard test problems confirm the theory presented

certificateless threshold cryptosystem secure against chosen ciphertext attack this paper proposes the first certificateless threshold decryption scheme which avoids both the single point of failure in the distributed networks and the inherent key escrow problem in identity based cryptosystem the scheme is secure against threshold chosen ciphertext attack it tolerates the type i adversary that can replace public key and the type ii adversary that has access to the system s master key the formal proof of security is presented in the random oracle model assuming some underlying problems closely related to the bilinear diffie hellman problem are computationally hard