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.

scholarly journals Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models

2018 ◽  
Vol 11 (4) ◽  
pp. 1497-1515 ◽  
Author(s):  
David J. Gardner ◽  
Jorge E. Guerra ◽  
François P. Hamon ◽  
Daniel R. Reynolds ◽  
Paul A. Ullrich ◽  
...  
Keyword(s):  
Acoustic Waves ◽  
Time Integration ◽  
Coupled System ◽  
Atmospheric Model ◽  
Step Size ◽  
Time Step ◽  
Solution Quality ◽  
Time Step Size ◽  
Runge Kutta ◽  
The Impact

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.

scholarly journals Implicit-explicit (IMEX) Runge-Kutta methods for non-hydrostatic atmospheric models

2017 ◽  
Author(s):  
David J. Gardner ◽  
Jorge E. Guerra ◽  
François P. Hamon ◽  
Daniel R. Reynolds ◽  
Paul A. Ullrich ◽  
...  
Keyword(s):  
Acoustic Waves ◽  
Time Integration ◽  
Coupled System ◽  
Atmospheric Model ◽  
Step Size ◽  
Time Step ◽  
Solution Quality ◽  
Time Step Size ◽  
Runge Kutta ◽  
The Impact

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work we investigate various implicit-explicit (IMEX) additive Runge-Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally implicit-vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored. The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy. Overall, the third order ARS343 and ARK324 methods performed the best, followed by the second order ARS232 and ARK232 methods.

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.

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 A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2284
Author(s):  
Endre Kovács ◽  
Ádám Nagy ◽  
Mahmoud Saleh
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Second Order ◽  
Stiff Systems ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Two Stage ◽  
Time Step Size ◽  
New Methods

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

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.

Traffic Flow Simulation through Parallel Processing

1996 ◽  
Vol 1566 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Anthony Theodore Chronopoulos ◽  
Gang Wang
Keyword(s):  
Numerical Methods ◽  
Traffic Flow ◽  
Traffic Simulation ◽  
Parallel Computer ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
The Past ◽  
Simulation Codes

Numerical methods for solving traffic flow continuum models have been studied and efficiently implemented in traffic simulation codes in the past. Explicit and implicit methods have been used in traffic simulation codes in the past. Implicit methods allow a much larger time step size than explicit methods to achieve the same accuracy. However, at each time step a nonlinear system must be solved. The Newton method, coupled with a linear iterative method (Orthomin), is used. The efficient implementation of explicit and implicit numerical methods for solving the high-order flow conservation traffic model on parallel computers was studied. Simulation tests were run with traffic data from an 18-mile freeway section in Minnesota on the nCUBE2 parallel computer. These tests gave the same accuracy as past tests, which were performed on one-processor computers, and the overall execution time was significantly reduced.

A Numerical Simulation of Swirling Two-Phase Flow in BWR Steam-Water Separator and Its Optimization

2003 ◽  
Author(s):  
Haruo Terasaka ◽  
Sensuke Shimizu ◽  
Minoru Kawahara
Keyword(s):  
Transient Flow ◽  
Fluid Model ◽  
Flow Analysis ◽  
Phase Flow ◽  
Step Size ◽  
Time Step ◽  
Two Phase ◽  
Time Step Size ◽  
Fully Implicit ◽  
Water Separator

An advanced numerical method based on the two-fluid model has been developed. The solution method presented here is an extension of the SIMPLEST scheme, a fully implicit scheme for single-phase flow analysis. It is robust and unconditionally stable, and therefore it enables us to use a very large time step size. This feature is suitable for steady and/or slow transient flow analyses. Furthermore, it enhances numerical stability during rapid transient calculations. By using this method, swirling gas-liquid flow in a steam-water separator of Boiling Water Reactors (BWRs) was calculated and the hydrodynamics characteristics were investigated for optimization.

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