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1.
Summary For the numerical solution of non-stiff semi-explicit differentialalgebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods. 相似文献
2.
Stephen L. Keeling 《BIT Numerical Mathematics》1989,29(1):91-109
Because of their potential for offering a computational speed-up when used on certain multiprocessor computers, implicit Runge-Kutta methods with a stability function having distinct poles are analyzed. These are calledmultiply implicit (MIRK) methods, and because of the so-calledorder reduction phenomenon, their poles are required to be real, i.e., only real MIRK's are considered. Specifically, it is proved that a necessary condition for aq-stage, real MIRK to beA-stable with maximal orderq+1 is thatq=1, 2, 3 or 5. Nevertheless, it is shown that for every positive integerq, there exists aq-stage, real MIRK which is stronglyA
0-stable with orderq+1, and for every evenq, there is aq-stage, real MIRK which isI-stable with orderq. Finally, some useful examples of algebraically stable real MIRK's are given.This work was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18107 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665-5225. 相似文献
3.
Hermann Brunner 《BIT Numerical Mathematics》1997,37(1):1-12
In this paper we analyze the attainable order ofm-stage implicit (collocation-based) Runge-Kutta methods for differential equations and Volterra integral equations of the
second kind with variable delay of the formqt (0<q<1). It will be shown that, in contrast to equations without delay, or equations with constant delay, collocation at the Gauss
(-Legendre) points will no longer yield the optimal (local) orderO(h
2m
).
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Research Grant OGP0009406). 相似文献
4.
Previously, the authors [9] classified various types of continuous explicit Runge-Kutta methods of order 5. Here, new lower bounds on the numbers of stages required for a sequence of continuous methods of increasing orders which are embedded in a continuouss-stage method of orderp are obtained. Carnicer [2] showed for each continuous explicit Runge-Kutta method of orderp in a mildly restricted family that at least 2p – 2 stages are required. Here, the same bound is established for all such methods of orderp.This research was supported by the Natural Sciences and Engineering Research Council of Canada, and the Information Technology Research Centre of Ontario. In addition, the second author was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica of Italy. 相似文献
5.
J. Schropp 《Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences》2013,19(2):263-271
We analyze Runge-Kutta discretizations applied to nonautonomous index 2 differential algebraic equations (DAEs) in semi-explicit form. It is shown that for half-explicit and projected Runge-Kutta methods there is an attractive invariant manifold for the discrete system which is close to the invariant manifold of the DAE. The proof combines reduction techniques to autonomou index 2 differential algebraic equations with some invariant manifold results of Schropp [9]. The results support the favourable behavior of these Runge-Kutta methods applied to index 2 DAEs for t = 0. 相似文献
6.
The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally
these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently
proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly
damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the
diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix.
The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences). 相似文献
7.
Unconditionally stable explicit methods for parabolic equations 总被引:2,自引:0,他引:2
E. Hairer 《Numerische Mathematik》1980,35(1):57-68
Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA
0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented. 相似文献
8.
Ch. Lubich 《BIT Numerical Mathematics》1991,31(3):545-550
Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems. 相似文献
9.
C. Bendtsen 《BIT Numerical Mathematics》1997,37(1):221-226
It is well known that high stage order is a desirable property for implicit Runge-Kutta methods. In this paper it is shown
that it is always possible to construct ans-stage IRK method with a given stability function and stage orders−1 if the stability function is an approximation to the exponential function of at least orders. It is further indicated how to construct such methods as well as in which cases the constructed methods will be stiffly
accurate. 相似文献
10.
Michael Müller 《BIT Numerical Mathematics》1992,32(4):676-688
The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable. 相似文献
11.
In quasistatic solid mechanics the initial boundary value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements. For the temporal discretization three different classes of Runge-Kutta methods are compared. These methods are diagonally implicit Runge-Kutta schemes (DIRK), linear implicit Runge-Kutta methods (Rosenbrock type methods) and half-explicit Runge-Kutta schemes (HERK). It will be shown that the application of half-explicit or linear-implicit Runge-Kutta methods can enormously reduce the computational time in particular situations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
12.
Equilibria of Runge-Kutta methods 总被引:2,自引:0,他引:2
Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4. 相似文献
13.
Apart from specific methods amenable to specific problems, symplectic Runge-Kutta methods are necessarily implicit. The aim
of this paper is to construct explicit Runge-Kutta methods which mimic symplectic ones as far as the linear growth of the
global error is concerned. Such method of orderp have to bepseudo-symplectic of pseudosymplecticness order2p, i.e. to preserve the symplectic form to within ⊗(h
2p
)-terms. Pseudo-symplecticness conditions are then derived and the effective construction of methods discussed. Finally, the
performances of the new methods are illustrated on several test problems. 相似文献
14.
Suchitra Gupta 《BIT Numerical Mathematics》1985,25(1):233-241
A class of finite difference schemes for the solution of a nonlinear system of first order differential equations with two point boundary conditions which shares properties with Runge-Kutta processes and gap schemes is discussed. The order conditions for the coefficients of these processes, techniques for reducing these order conditions in number and the symmetry conditions are given. A symmetricA-stable eight order process which has second, fourth and sixth orderA-stable processes embedded in it is given as an example.Research supported in part by the United States Air Force under contract AFOSR-89-0383. 相似文献
15.
Aubry and Chartier introduced (1998) the concept of pseudo-symplecticness in order to construct explicit Runge-Kutta methods,
which mimic symplectic ones. Of particular interest are methods of order (p, 2p), i.e., of orderp and pseudo-symplecticness order 2p, for which the growth of the global error remains linear. The aim of this note is to show that the lower bound for the minimal
number of stages can be achieved forp=4 andp=5. 相似文献
16.
The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations 总被引:8,自引:0,他引:8
Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of
neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable.
Received February 10, 1997 / Revised version received January 5, 1998 相似文献
17.
《Journal of Computational and Applied Mathematics》1997,87(1):147-167
Mono-implicit Runge-Kutta methods can be used to generate implicit Runge-Kutta-Nyström (IRKN) methods for the numerical solution of systems of second-order differential equations. The paper is concerned with the investigation of the conditions to be fulfilled by the mono-implicit Runge-Kutta (MIRK) method in order to generate a mono-implicit Runge-Kutta-Nyström method (MIRKN) that is P-stable. One of the main theoretical results is the property that MIRK methods (in standard form) cannot generate MIRKN methods (in standard form) of order greater than 4. Many examples of MIRKN methods generated by MIRK methods are presented. 相似文献
18.
To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is
very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard
Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize.
We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with
respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the
estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori
step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems
of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and
nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision,
while the aggregative Taylor series method achieves the best computational time.
The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of
ODEs describing the optimal flight of a spacecraft from the Earth to the Moon.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical
Systems and Optimization, 2005. 相似文献
19.
A wide class of discretisation methods for ordinary differential equations is introduced and a new concept of consistency, called optimal consistency, is defined. This permits convergence of order exactlyp (that is, two sided error bounds) when the method is optimally consistent of orderp. This is then related to the minimal and optimal stability functionals introduced by Spijker and Albrecht, and a new algebraic criterion is given for a discretisation method consistent of orderp to be convergent of orderp + 1. Finally it is shown that the original motivation for the idea of optimal consistency arises from discretisation methods for Volterra integral equations. 相似文献
20.
In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods,
the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular,
methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present
in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these
stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation.
In particular, it will be shown that if ans-stage SRK contains Stratonovich integrals up to orderp then the strong order of the SRK cannot exceed min{(p+1)/2, (s−1)/2},p≥2,s≥3 or 1 ifp=1. 相似文献