Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

PARALLEL PREDICTOR-CORRECTOR METHODS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS II

1999 ◽  
Vol 10 (01) ◽  
pp. 135-145
Author(s):  
A. M. MAZZONE
Keyword(s):  
Parallel Computing ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Methods ◽  
Performance Comparison ◽  
Molecular Dynamics Calculations ◽  
The Subject ◽  
The Stability ◽  
Predictor Corrector ◽  
Serial Algorithms

This work presents parallel multistep methods for the solution of ordinary differential equations. The characteristic of parallel computing is that there is a "front" and the computation at points ahead of the front depends only on information behind it. This requires a resetting of serial algorithms and may also lead to numerical errors and instabilities. The analysis of positive and negative aspects of parallel computing is the subject of this paper. Some of the methods presented below are uncommon in the literature on mathematical computing. Others have been elaborated for this study on the basis of the traditional Adams-Bashforth multistep methods. A performance comparison of the methods is made by numerical testing in molecular dynamics calculations. The increase of the number of processors m appears to seriously deteriorate the stability of the calculations and the use of m larger than 2 seems impractical.

AN ADVANCED ADAPTIVE FINITE ELEMENT CODE APPLIED FOR COATING-SUBSTRATE SIMULATION

2011 ◽  
Vol 03 (01n02) ◽  
pp. 91-107 ◽  
Author(s):  
JÜRGEN LEOPOLD ◽  
KATRIN HELLER ◽  
ARNDT MEYER ◽  
REINER WOHLGEMUTH
Keyword(s):  
Finite Element ◽  
Fe Simulation ◽  
Scale Up ◽  
Computational Time ◽  
Adaptive Finite Element ◽  
Workpiece Surface ◽  
Coated Tools ◽  
Geometrical Simulation ◽  
The Stability ◽  
And Storage

The stability of coating-substrate systems influences the chip formation and the surface integrity of the new generated workpiece surface, too. Using finite element (FE) simulation, deformations, strains and stresses in coated tools, caused by external and internal loads, can be computed on a microscopic scale. Since both, the whole macroscopic tool (in mm-scale) and the microscopic coating layers (in μm-scale up to nm-scale) must be included in the same geometrical simulation model, graded high-resolution FE meshes must be used. Nevertheless, the number of nodes in the 3D computational FE grid reaches some millions, leading to large computational time and storage requirements. For this reason, an advanced adaptive finite element (AAFEM) software has been developed and used for the simulation.

scholarly journals IL-GLOBO (1.0) – integrated Lagrangian particle model and Eulerian general circulation model GLOBO: development of the vertical diffusion module

2014 ◽  
Vol 7 (5) ◽  
pp. 2181-2191 ◽  
Author(s):  
D. Rossi ◽  
A. Maurizi
Keyword(s):  
General Circulation Model ◽  
General Circulation ◽  
Spline Interpolation ◽  
Circulation Model ◽  
Linear Interpolation ◽  
Computational Time ◽  
Integration Time ◽  
Particle Model ◽  
Vertical Diffusion ◽  
Time Step

Abstract. The development and validation of the vertical diffusion module of IL-GLOBO, a Lagrangian transport model coupled online with the Eulerian general circulation model GLOBO, is described. The module simulates the effects of turbulence on particle motion by means of a Lagrangian stochastic model (LSM) consistently with the turbulent diffusion equation used in GLOBO. The implemented LSM integrates particle trajectories, using the native σ-hybrid coordinates of the Eulerian component, and fulfils the well-mixed condition (WMC) in the general case of a variable density profile. The module is validated through a series of 1-D offline numerical experiments by assessing its accuracy in maintaining an initially well-mixed distribution in the vertical. A dynamical time-step selection algorithm with constraints related to the shape of the diffusion coefficient profile is developed and discussed. Finally, the skills of a linear interpolation and a modified Akima spline interpolation method are compared, showing that both satisfy the WMC with significant differences in computational time. A preliminary run of the fully integrated 3-D model confirms the result only for the Akima interpolation scheme while the linear interpolation does not satisfy the WMC with a reasonable choice of the minimum integration time step.

Numerical Determination of the Response of a Linear Vibration System With a Singular Mass Matrix

1972 ◽  
Vol 94 (1) ◽  
pp. 64-69
Author(s):  
K. D. Willmert
Keyword(s):  
Multistep Methods ◽  
Mass Matrix ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Vibrational System ◽  
Wide Range ◽  
Mass Matrices ◽  
The Stability ◽  
The Many

In numerically determining the response of a linear second-order multidegree-of-freedom vibrational system subjected to a general excitation, the common approach of applying one of the many multistep methods of numerical analysis (e.g., Milne-Simpson, Adams-Bashforth, etc.) leads ultimately to the solution of a system of linear equations. However, when the mass matrix of the original vibrational system is singular, the coefficient matrix of the system of equations also becomes singular and thus the response cannot be determined. Presented is a means of applying these multistep methods to vibrational systems which results in a method that is capable of obtaining the response independent of the singularity of the mass matrix. This technique is particularly useful in optimization where the values of the parameters of the system are unknown in advance, and thus the method of determining the response must be applicable for a wide range of values of the parameters. In the development and investigation of this technique, the causes of the stability problems which develop from the application of multistep methods to systems with nearly singular mass matrices become apparent.

scholarly journals On the stability of implicit-explicit linear multistep methods

1997 ◽  
Vol 25 (2-3) ◽  
pp. 193-205 ◽  
Author(s):  
J. Frank ◽  
W. Hundsdorfer ◽  
J.G. Verwer
Keyword(s):  
Multistep Methods ◽  
Linear Multistep Methods ◽  
The Stability

Stability and Bifurcation of Nonlinear Bearing-Flexible Rotor System with a Single Disk

2010 ◽  
Vol 148-149 ◽  
pp. 141-146
Author(s):  
Di Hei ◽  
Yong Fang Zhang ◽  
Mei Ru Zheng ◽  
Liang Jia ◽  
Yan Jun Lu
Keyword(s):  
Nonlinear Dynamic ◽  
Degrees Of Freedom ◽  
Rotor System ◽  
Dynamic Responses ◽  
Flexible Rotor ◽  
Runge Kutta Method ◽  
Stability And Bifurcation ◽  
Single Disk ◽  
The Stability ◽  
Predictor Corrector

Dynamic model and equation of a nonlinear flexible rotor-bearing system are established based on rotor dynamics. A local iteration method consisting of improved Wilson-θ method, predictor-corrector mechanism and Newton-Raphson method is proposed to calculate nonlinear dynamic responses. By the proposed method, the iterations are only executed on nonlinear degrees of freedom. The proposed method has higher efficiency than Runge-Kutta method, so the proposed method improves calculation efficiency and saves computing cost greatly. Taking the system parameter ‘s’ of flexible rotor as the control parameter, nonlinear dynamic responses of rotor system are obtained by the proposed method. The stability and bifurcation type of periodic responses are determined by Floquet theory and a Poincaré map. The numerical results reveal periodic, quasi-periodic, period-5, jump solutions of rich and complex nonlinear behaviors of the system.

A numerical algorithm for solving robot inverse kinematics

Robotica ◽  
1989 ◽  
Vol 7 (2) ◽  
pp. 159-164 ◽  
Author(s):  
K. C. Gupta ◽  
V. K. Singh
Keyword(s):  
Inverse Kinematics ◽  
Computational Time ◽  
Robot Kinematics ◽  
Convergence Criteria ◽  
Euler Parameters ◽  
Rotational Error ◽  
Variable Step ◽  
Zero Reference ◽  
Step Algorithm ◽  
Predictor Corrector

SUMMARYAn extension of the inverse kinematics algorithm by Gupta and Kazerounian is presented. The robot kinematics is formulated by using the Zero Reference Position Method. Euler parameters and the related vector forms of the spatial rotation concatenation have been used to improve the efficiency of the velocity Jacobian computation. The joint rates are formally integrated by using a modified predictor-corrector method particularized to robot inverse kinematics – it is a strict descending, p(1)c(0 – n), variable step algorithm. The definitions of the rotational error and overall error measure have been revised. Depending upon the convergence criteria used, these modifications reduce the overall computational time by 20%.

Time-Dependent, Three Dimensional Nodal Transport Code Development Based on Unstructured Mesh

2014 ◽  
Author(s):  
Mingtao He ◽  
Hongchun Wu ◽  
Liangzhi Cao ◽  
Youqi Zheng ◽  
ShengCheng Zhou
Keyword(s):  
Direct Method ◽  
Three Dimensional ◽  
Computational Time ◽  
Neutron Kinetics ◽  
Nodal Method ◽  
Transport Code ◽  
Geometry Configuration ◽  
Nodal Transport ◽  
Complicated Geometry ◽  
Predictor Corrector

A space-time nodal transport code, DAISY, was developed to evaluate dynamic neutron behavior in innovative nuclear system. The steady transport process is based on an arbitrary triangles-z mesh nodal method which can treat complicated geometry configuration with enough precision and acceptable calculated quantity. This code employs the improved quasi-static method for neutron kinetics with a predictor-corrector scheme to improve computational efficiency. The direct method and the point approximation for neutron kinetics are also implemented into DAISY to evaluate the precision and efficiency of this predictor-corrector scheme. This code was verified by several transient benchmarks. It shows that the predictor-corrector scheme in DAISY can greatly reduce the computational time with enough precision.

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