Integration of stiff mechanical systems by Runge-Kutta methods

The numerical integration of stiff mechanical systems is studied in which a strong potential forces the motion to remain close to a manifold. The equations of motion are written as a singular singular perturbation problem with a small stiffness parameter . Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than are shown to approximate smooth solutions, and precise error estimates are derived. As 0, Runge-Kutta solutions of the stiff system converge to Runge-Kutta solutions of the associated constrained system formulated as a differential-algebraic equation of index 3. Standard software for stiff initial-value problems does not work satisfactorily on the stiff systems considered here. The reasons for this failure are explained, and remedies are proposed.This work was supported in part by the Austrian Science Foundation, grant P8443-PHY.     

Explicit methods for mildly stiff oscillatory systems

We discuss error control for explicit methods when the stepsize is bounded by stability on the imaginary axis. Our main result is a formulation of a condition on the estimator of the local error which prevents the fast components to exceed the prescribed error tolerance. A PECE Adams method of 4th order accuracy is proposed for mildly stiff oscillatory systems. For comparison we also discuss embedded Runga-Kutta methods.Partially supported by the Office of Naval Research N00014-90-J-1382     

Relaxed Newton-like methods for stiff differential systems

Newton-like methods are commonly used to solve the nonlinear equations arising in the numerical solution of stiff differential equations. We show that easily calculable relaxation factors may be used to improve the convergence properties of such methods. The technique is also applicable when partitioning methods are used.     

New second derivative multistep methods for stiff systems

In this paper we present details of a new class of second derivative multistep methods. Stability analysis is discussed and an improvement in stability region is obtained. With a simple modification we take advantage of calling for the solution of algebraic equations with the same coefficient matrix in each step. Moreover, using IOM to solve the resulting linear systems, the coefficient matrix is not needed explicitly. Some numerical experiments and comparison with several popular codes are given, showing strong superiority of this new class of methods.     

非线性刚性微分-代数系统的波形松弛离散法

研究基于Runge-Kutta方法的波形松弛离散过程,得到新的刚性微分-代数系统的收敛理论,及该类系统解的存在性和惟一性,并用具体算例测试该理论的有效实用性.     

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi??s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s?1 in the singly-implicit case with arbitrary step sizes and s??5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.     

Rational-fractional methods for solving stiff systems of differential equations

This paper proposes new numerical methods for solving stiff systems of first-order differential equations not resolved with respect to the derivative. These methods are based on rational-fractional approximations of the vector-valued function of solution of the system considered. The authors study the stability of the constructed methods of arbitrary finite order of accuracy. Analysis of the results of experimental studies of these methods by test examples of various types confirms their efficiency. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 203–208, 2006.     

New efficient second derivative multistep methods for stiff systems

The aim of this paper is to select from the large family of possible general linear methods, just a single class which has considerable potential for efficient implementation. This class has possible applications depending on stiff nature of a problem to be performed. A special class of second derivative multistep method (SDMM) is derived. The stability analysis of this class which is depending on free parameters is discussed. The stability regions are plotted for certain choices of parameters. A good comparison between the results of this class and the results due to Gear and Enright is recommended during some numerical tests.     

Convex structures induced by Chebyshev systems

Recent developments have clarified that some tools of Convex Geometry are closely related to separation theorems obtained in the field of Functional Inequalities. This phenomenon has motivated the investigation of convex structures induced by Chebyshev systems. The present note characterizes such a possible structure, completely describing its combinatorial invariants.     

One-step methods of hermite type for numerical integration of stiff systems

One-step methods of Hermite type with coefficients equal to the derivatives of Laguerre polynomials at certain points are considered. The methods areA-stable of order 1, 2, 3, 5 and for order higher than 5 they are “nearly”A-stable. Used with special linear problems the matrix inversion turns out to be simple.     

Rational Runge-Kutta methods are not suitable for stiff systems of ODEs

The present paper shows that rational RK-methods are not very appropriate to solve stiff differential equations. The CA₀-stability (i.e. componentwise contractivity) is defined and the non-existence of CA₀-stable rational RK-methods is demonstrated. Furthermore it is shown that the stepsizes which can be expected when solving a stiff differential system with a rational or with an explicit linear RK-method are of the same order of magnitude.     

Partitioned adaptive Runge-Kutta methods for the solution of nonstiff and stiff systems

Summary For the numerical solution of initial value problems of ordinary differential equations partitioned adaptive Runge-Kutta methods are studied. These methods consist of an adaptive Runge-Kutta methods for the treatment of a stiff system and a corresponding explicit Runge-Kutta method for a nonstiff system. First we modify the theory of Butcher series for partitioned adaptive Runge-Kutta methods. We show that for any explicit Runge-Kutta method there exists a translation invariant partitoned adaptive Runge-Kutta method of the same order. Secondly we derive a special translaton invariant partitioned adaptive Runge-Kutta method of order 3. An automatic stiffness detection and a stepsize control basing on Richardson-extrapolation are performed. Extensive tests and comparisons with the partitioned RKF4RW-algorithm from Rentrop [16] and the partitioned algorithm LSODA from Hindmarsh [9] and Petzold [15] show that the partitoned adaptive Runge-Kutta algorithm works reliable and gives good numericals results. Furthermore these tests show that the automatic stiffness detection in this algorithm is effective.     

On one class of numerical methods for solving stiff systems of differential equations

The paper is concerned with a modification of a Cowell-type method: at each step the solution is evaluated at several points and only some first of these values are retained. Stability of these methods is examined. In particular, among the methods of this group we single out one method that has the fourth order of accuracy and is stable for stiff systems with any choice of integration step.     

A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods (ABC-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in ABC-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. ABC-schemes are A- and L-stable methods of the second order, but there are ABC-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of ABC-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.     

A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

Two efficient third-and fourth-order processes for solving the initial value problem for ordinary differential equations are studied. Both are A-stable and so recommended for stiff systems. An economic and efficient way of step-size control is given for each of them. Numerical examples are considered.