Unconditional Stability in Numerical Time Integration Methods

1973 ◽  
Vol 40 (2) ◽  
pp. 417-421 ◽  
Author(s):  
R. D. Krieg
Keyword(s):  
Time Integration ◽  
Step Size ◽  
Time Step ◽  
Central Difference ◽  
Integration Methods ◽  
Time Step Size ◽  
Time Integration Method ◽  
Time Integration Methods ◽  
Numerical Time Integration ◽  
Difference Time

Methods of numerical time integration of the equation M¯q¨ + K¯q = f are examined in this paper. A particular class of explicit time integration methods is defined and this class is searched for an unconditionally stable method. The class is found to contain no such method and, furthermore, is found to contain no method with a larger stable time step size than that characterized by the simple central difference time integration method.

scholarly journals Implementation of multirate time integration methods for air pollution modelling

2012 ◽  
Vol 5 (6) ◽  
pp. 1395-1405 ◽  
Author(s):  
M. Schlegel ◽  
O. Knoth ◽  
M. Arnold ◽  
R. Wolke
Keyword(s):  
Time Integration ◽  
Efficient Implementation ◽  
Step Size ◽  
Time Step ◽  
Worst Case ◽  
Integration Methods ◽  
Integration Schemes ◽  
Numerical Effort ◽  
Multirate Time Integration ◽  
Time Integration Methods

Abstract. Explicit time integration methods are characterised by a small numerical effort per time step. In the application to multiscale problems in atmospheric modelling, this benefit is often more than compensated by stability problems and step size restrictions resulting from stiff chemical reaction terms and from a locally varying Courant-Friedrichs-Lewy (CFL) condition for the advection terms. Splitting methods may be applied to efficiently combine implicit and explicit methods (IMEX splitting). Complementarily multirate time integration schemes allow for a local adaptation of the time step size to the grid size. In combination, these approaches lead to schemes which are efficient in terms of evaluations of the right-hand side. Special challenges arise when these methods are to be implemented. For an efficient implementation, it is crucial to locate and exploit redundancies. Furthermore, the more complex programme flow may lead to computational overhead which, in the worst case, more than compensates the theoretical gain in efficiency. We present a general splitting approach which allows both for IMEX splittings and for local time step adaptation. The main focus is on an efficient implementation of this approach for parallel computation on computer clusters.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
Weicheng Huang ◽  
Mohammad Khalid Jawed
Keyword(s):  
Energy Dissipation ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Step ◽  
Elastic Rods ◽  
Computationally Efficient ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Integration Scheme

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

A Precise and Stiffly Stable Time Integration Method for Vibration Equations

2001 ◽  
Author(s):  
Takeshi Fujikawa ◽  
Etsujiro Imanishi
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Quadratic Function ◽  
Time Step ◽  
Integration Algorithm ◽  
Time Integration Method ◽  
Time Integration Algorithm ◽  
Time Integration Methods ◽  
Motion Problems

Abstract A method of time integration algorithm is presented for solving stiff vibration and motion problems. It is absolutely stable, numerically dissipative, and much accurate than other dissipative time integration methods. It achieves high-frequency dissipation, while minimizing unwanted low-frequency dissipation. In this method change of acceleration during time step is expressed as quadratic function including some parameters, whose appropriate values are determined through numerical investigation. Two calculation examples are demonstrated to show the usefulness of this method.

scholarly journals Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

2010 ◽  
Vol 138 (8) ◽  
pp. 3333-3341 ◽  
Author(s):  
Katherine J. Evans ◽  
Mark A. Taylor ◽  
John B. Drake
Keyword(s):  
Shallow Water Equation ◽  
Time Integration ◽  
Atmospheric Model ◽  
Test Cases ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Fully Implicit ◽  
Stability Limits

Abstract A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Direct Numerical Simulation of the Interaction of an Ultra Short-Pulsed Intense Laser With a H2+ Molecule

2008 ◽  
Author(s):  
Y.-M. Lee ◽  
J.-S. Wu ◽  
T.-F. Jiang ◽  
Y.-S. Chen
Keyword(s):  
Domain Decomposition Method ◽  
Time Integration ◽  
Three Dimensional ◽  
Time Algorithm ◽  
Intense Laser ◽  
Angle Of Incidence ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
H2 Molecule

In this paper, interactions of a linearly polarized ultra short-pulsed intense laser with a single H2+ molecule at various angles of incidence are studied by directly solving the time-dependent three-dimensional Schrodinger equation (TDSE), assuming Born-Oppenheimer approximation. An explicit stagger-time algorithm is employed for time integration of the TDSE, in which the real and imaginary parts of the wave function are defined at alternative times, while a cell-centered finite-volume method is utilized for spatial discretization of the TDSE on Cartesian grids. The TDSE solver is then parallelized using domain decomposition method on distributed memory machines by applying a multi-level graph-partitioning technique. The solver is applied to simulate laser-molecular interaction with test conditions including: laser intensity of 0.5*1014 W/cm2, wavelength of 800 nm, three pulses in time, angle of incidence of 0–90° and inter-nuclear distance of 2 a.u.. Simulation conditions include 4 million hexahedral cells, 90 a.u. long in z direction, and time-step size of 0.005 a.u.. Ionization rates, harmonic spectra and instantaneous distribution of electron densities are then obtained from the solution of the TDSE. Future possible extension of the present method is also outlined at the end of this paper.

scholarly journals Acceleration of the IMplicit–EXplicit nonhydrostatic unified model of the atmosphere on manycore processors

2017 ◽  
Vol 33 (2) ◽  
pp. 242-267 ◽  
Author(s):  
Daniel S Abdi ◽  
Francis X Giraldo ◽  
Emil M Constantinescu ◽  
Lester E Carr ◽  
Lucas C Wilcox ◽  
...  
Keyword(s):  
Weather Prediction ◽  
Time Integration ◽  
Explicit Methods ◽  
Benchmark Problems ◽  
Graphic Processing Units ◽  
Courant Number ◽  
Time Step ◽  
Integration Methods ◽  
Manycore Processors ◽  
Time Integration Methods

We present the acceleration of an IMplicit–EXplicit (IMEX) nonhydrostatic atmospheric model on manycore processors such as graphic processing units (GPUs) and Intel’s Many Integrated Core (MIC) architecture. IMEX time integration methods sidestep the constraint imposed by the Courant–Friedrichs–Lewy condition on explicit methods through corrective implicit solves within each time step. In this work, we implement and evaluate the performance of IMEX on manycore processors relative to explicit methods. Using 3D-IMEX at Courant number C = 15, we obtained a speedup of about 4× relative to an explicit time stepping method run with the maximum allowable C = 1. Moreover, the unconditional stability of IMEX with respect to the fast waves means the speedup can increase significantly with the Courant number as long as the accuracy of the resulting solution is acceptable. We show a speedup of 100× at C = 150 using 1D-IMEX to demonstrate this point. Several improvements on the IMEX procedure were necessary in order to outperform our results with explicit methods: (a) reducing the number of degrees of freedom of the IMEX formulation by forming the Schur complement, (b) formulating a horizontally explicit vertically implicit 1D-IMEX scheme that has a lower workload and better scalability than 3D-IMEX, (c) using high-order polynomial preconditioners to reduce the condition number of the resulting system, and (d) using a direct solver for the 1D-IMEX method by performing and storing LU factorizations once to obtain a constant cost for any Courant number. Without all of these improvements, explicit time integration methods turned out to be difficult to beat. We discuss in detail the IMEX infrastructure required for formulating and implementing efficient methods on manycore processors. Several parametric studies are conducted to demonstrate the gain from each of the abovementioned improvements. Finally, we validate our results with standard benchmark problems in numerical weather prediction and evaluate the performance and scalability of the IMEX method using up to 4192 GPUs and 16 Knights Landing processors.

Modeling the damped dynamic behavior of a flexible pendulum

2019 ◽  
Vol 54 (2) ◽  
pp. 116-129 ◽  
Author(s):  
Roberto Ortega ◽  
Geraldine Farías ◽  
Marcela Cruchaga ◽  
Matías Rivero ◽  
Mariano Vázquez ◽  
...  
Keyword(s):  
High Speed ◽  
Time Integration ◽  
Integration Scheme ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Rayleigh Damping ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
Damping Model

The focus of this work is on the computational modeling of a pendulum made of a hyperelastic material and the corresponding experimental validation with the aim of contributing to the study of a material commonly used in seismic absorber devices. From the proposed dynamics experiment, the motion of the pendulum is recorded using a high-speed camera. The evolution of the pendulum’s positions is recovered using a capturing motion technique by tracking markers. The simulation of the problem is developed in the framework of a parallel multi-physics code. Particular emphasis is placed on the analysis of the Newmark integration scheme and the use of Rayleigh damping model. In particular, the time step size effect is analyzed. A strong time step size dependency is obtained for dissipative time integration schemes, while the Rayleigh damping formulation without time integration dissipation shows time step–independent results when convergence is achieved.

Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem

2011 ◽  
Vol 10 (4) ◽  
pp. 844-866 ◽  
Author(s):  
Jingyan Yue ◽  
Guangwei Yuan
Keyword(s):  
Iterative Method ◽  
Time Integration ◽  
Step Size ◽  
Time Step ◽  
Radiation Diffusion ◽  
Reaction Diffusion Systems ◽  
Time Step Size ◽  
Step Control ◽  
Equilibrium Radiation ◽  
Non Equilibrium

AbstractFor a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

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