Showing papers on "L-stability published in 1994"
TL;DR: A new class of methods is determined, which is different from Runge–Kutta methods but closely related to them, and which combines its high order of accuracy and unconditional contractivity in the maximum norm.
Abstract: Contractivity properties of Runge–Kutta methods are analyzed, with suitable interpolation implemented using waveform relaxation strategy for systems of ordinary differential equations that are dissipative in the maximum norm. In general, this type of implementation, which is quite appropriate in a parallel computing environment, improves the stability properties of Runge–Kutta methods. As a result of this analysis, a new class of methods is determined, which is different from Runge–Kutta methods but closely related to them, and which combines its high order of accuracy and unconditional contractivity in the maximum norm. This is not possible for classical Runge–Kutta methods.
43 citations
TL;DR: In this article, a new stiff ordinary differential equation solver was devised that separates the unknown variables into a fast group and a slow group, and the fast variables are solved using the implicit backward-differentiation formulas but with a Jacobian of much smaller dimension than that of the original stiff system.
36 citations
TL;DR: In this paper, the authors considered the one-block methods designed by Sommeijer, Couzy, and van der Houwen for the purpose of solving ODEs on a parallel computer.
Abstract: In this contribution, the author considers the one-block methods designed by Sommeijer, Couzy, and van der Houwen for the purpose of solving ordinary differential equations (ODEs) on a parallel computer. The author also derives a new set of order conditions, studies the stability, and exhibits a new class of parallel methods which contains L-stable schemes up to order eleven.
27 citations
TL;DR: A new method of calculating closed-chain dynamics is proposed, based on the systematic elimination of variables that are both redundant and that may adversely affect the computations, which produces numerically stable solutions of the differential equations of motion.
Abstract: A common approach to formulating the dynamics of closed-chain mechanisms requires finding the forces of constraint at the loop closures. However, there are indications that this approach leads both to ill conditioned systems that must be inverted and to numerically unstable differential equations of motion. We derive a sufficient condition for ill conditioning of augmented dynamical systems/spl minus/that the mechanism's trajectory passes through, or very near, a kinematic singularity. In singular regions the equations of motion are also numerically stiff, and they frequently require special numerical methods for computer solution. We propose a new method of calculating closed-chain dynamics, based on the systematic elimination of variables that are both redundant and that may adversely affect the computations. This approach produces numerically stable solutions of the differential equations of motion, and the equations are apparently much less stiff than the equations produced by the traditional force-closure approach. >
21 citations
TL;DR: This paper investigates the situation, explaining some surprising time step selection behavior produced by the standard control mechanism, and proposes a new control mechanism which attempts to find and use a “stability” stepsize.
16 citations
TL;DR: The resulting scheme can be regarded as a parallel singly diagonally implicit Runge–Kutta (PDIRK) method, which requires effectively the solution of two implicit relations per step on a two-processor computer.
10 citations
TL;DR: In this paper, the ordinary differential equations of Newtonian dynamics are used in atomic simulations with the method of molecular dynamics, and the basic issues are surveyed and standard algorithms are described.
Abstract: The ordinary differential equations of Newtonian dynamics are used in atomic simulations with the method of molecular dynamics. The basic issues are surveyed and standard algorithms are described. Several algorithmic variants are discussed. Some advanced ideas relating to parallel computation are considered.
9 citations
TL;DR: In this paper, methods for the direct determination of stationary and periodic solutions of systems whose behavior is described by a set of ordinary differential equations, and the continuation of these solutions if a parameter is varied, are presented.
Abstract: Methods for the direct determination of stationary and periodic solutions of systems whose behaviour is described by a set of ordinary differential equations, and the continuation of these solutions if a parameter is varied, are presented. The realization of these methods in a program for flexible multibody systems is discussed, which requires, besides the determination of the equations of motion, the determination of the linearized equations and the sensitivity of the equations with respect to parameter variations. The methods are applied to an elastic rotor with mass eccentricity and a slider-crank mechanism with a flexible connecting rod.
7 citations
TL;DR: In this article, an algebraic characterization of I-, A-, and B-stability of Runge-Kutta methods is deduced without any assumption on the runge-kutta method.
Abstract: Important stability concepts for Runge–Kutta methods are I-, A-, and B-stability. For these properties there exist very similar algebraic characterizations. The characterization of B-stability is known for S-irreducible methods. In this paper, an algebraic characterization of I-stability and A-stability related to the coefficients of the method is deduced without any assumption on the Runge–Kutta methods. The corresponding linear dynamic system and its transfer function is considered. The positive real lemma characterizes the passivity of the system or equivalently the positive realness of the transfer function by the Lyapunov equation. Dropping the assumption of controllability and observability a generalization is possible using the Kalman canonical decomposition. Interpreting the modified stability function of a Runge–Kutta method as the transfer function, the positive real lemma yields a complete algebraic characterization of A-stable Runge–Kutta methods.
5 citations
TL;DR: The proposed construction is constructed for a special classes of stiff ODE, those of the form y'(t)=A(t)y(t)+g(t), which has a high efficiency in the implementation on a dis tributed memory multiprocessor when the ODEs function has many components.
Abstract: The problem associated with the stiff ordinary differential equa tion (ODE) systems in parallel processing is that the calculus can not be started simultaneously on many processors with an explicit formula. The proposed al gorithm is constructed for a special classes of stiff ODE, those of the form y'(t)=A(t)y(t)+g(t). It has a high efficiency in the implementation on a dis tributed memory multiprocessor when the ODEs function has many components. The approximation error is equal to that produced by the analogous sequential algorithm.
TL;DR: In this article, a new method of error control for the purpose of automatic choice of stepsize has been developed, where the error criterion is imposed on a suitable ellipsoidal norm of the vector of errors, the growth of global error is linear.
Abstract: The truncation error in the One-Step-Predictor-Corrector (OSPC) method of integrating a stiff system of Ordinary Differential Equations (ODE's) has been obtained. A new method of error-control for the purpose of automatic choice of stepsize has been developed. It is shown that if the error criterion is imposed on a suitable ellipsoidal norm of the vector of errors, the growth of global error is linear. Numerical tests show that this method of error-control leads to more accurate solutions than the conventional method which uses only a scalar error for controlling the time-step.
TL;DR: Parallel computers have changed the scene and have motivated us to design parallel iteration methods for solving the implicit systems in such a way that the resulting methods become efficient step-by-step methods for integrating stiff initial-value problems.
Abstract: From a theoretical point of view, Runge-Kutta methods of collocation type belong to the most attractive step-by-step methods for integrating stiff problems. These methods combine excellent stability features with the property of superconvergence at the step points. Like the initial-value problem itself, they only need the given initial value without requiring additional starting values, and therefore, are a natural discretization of the initial-value problem. On the other hand, from a practical point of view, these methods have the drawback of requiring in each step the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-value problem, respectively. In contrast, linear multistep methods, the main competitor of Runge-Kutta methods, require the solution of systems of dimension d. However, parallel computers have changed the scene and have motivated us to design parallel iteration methods for solving the implicit systems in such a way that the resulting methods become efficient step-by-step methods for integrating stiff initial-value problems.
TL;DR: Numerical examples show some essential qualitative requirements for numerical methods that are designed for stiff problems are very reliable, have good consistency and may be recommended for a wide range of problems.
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