scholarly journals Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1342
Author(s):  
Yu-Cheng Shen ◽  
Chia-Liang Lin ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Additional Property ◽  
Truncation Errors ◽  
Efficient Performance ◽  
Wide Range ◽  
Orbital Problems ◽  
Runge Kutta ◽  
Classical Orbits

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order.

scholarly journals Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

2021 ◽  
Author(s):  
Hendrik Ranocha ◽  
Lisandro Dalcin ◽  
Matteo Parsani ◽  
David I. Ketcheson
Keyword(s):  
Fluid Dynamics ◽  
Error Control ◽  
Time Integration ◽  
Size Control ◽  
Spectral Element ◽  
Step Size ◽  
Step Size Control ◽  
Wide Range ◽  
Cfd Applications ◽  
Runge Kutta

AbstractWe develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

Neville's and Romberg's processes: a fresh appraisal with extensions

1968 ◽  
Vol 263 (1144) ◽  
pp. 525-562 ◽  
Keyword(s):  
Power Series ◽  
Truncation Error ◽  
Point Of View ◽  
The Other ◽  
Matrix Formulation ◽  
Rounding Errors ◽  
Numerical Examples ◽  
Truncation Errors ◽  
Wide Range ◽  
Romberg Integration

In this paper Neville’s process for the repetitive linear combination of numerical estimates is re-examined and exhibited as a process for term-by-term elimination of error, expressed as a power series; this point of view immediately suggests a wide range of applications—other than interpolation, for which the process was originally developed, and which is barely mentioned in this paper—for example, to the evaluation of finite or infinite integrals in one or more variables, to the evaluation of sums, etc. A matrix formulation is also developed, suggesting further extensions, for example, to the evaluation of limits, derivatives, sums of series with alternating signs, and so on. It is seen also that Neville’s process may be readily applied in Romberg Integration; each suggests extensions of the other. Several numerical examples exhibit various applications, and are accompanied by comments on the behaviour of truncation and rounding errors as exhibited in each Neville tableau, to show how these provide evidence of progress in the improvement of the approximation, and internal numerical evidence of the nature of the truncation error. A fuller and more connected account of the behaviour of truncation errors and rounding errors is given in a later section, and suggestions are also made for choosing suitable specific original estimates, i.e. for choosing suitable tabular arguments in the elimination variable, in order to produce results as precise and accurate as possible.

Analysis and Implementation of Kerberos Protocol in Hybrid Cloud Computing Environments

2021 ◽  
Vol 39 (1B) ◽  
pp. 41-52
Author(s):  
Turkan A. Khaleel
Keyword(s):  
Cloud Computing ◽  
Hybrid Cloud ◽  
Computing Environment ◽  
Network Efficiency ◽  
Cloud Computing Environment ◽  
Computing Services ◽  
Efficient Performance ◽  
Wide Range ◽  
Before And After ◽  
Cloud Computing Services

The concept of cloud computing has recently changed how hardware, software, and information are handled. However, security challenges and credibility requirements have never changed and may have increased. Protecting cloud computing and providing security for its resources and users is one of the critical challenges. As a result, most users are afraid to use their resources, because many security problems must be met. For example, authentication and reliability are major security constraints and must be provided in a cloud computing environment. There is a wide range of authentication protocols in use, but the researcher has recommended the Kerberos protocol to represent and test it in a complex environment such as a mixed cloud environment. A model has been developed to implement Kerberos authentication in a hybrid cloud computing environment to securely access the cloud computing services provided. This model is represented using the OPNET Modeler 14.5 simulation system. The network efficiency was measured before and after the hacker. Findings presented in this research are supporting the ability of the Kerberos protocol to prevent illegal access to cloud computing services, whether from within the private cloud or the public cloud. While maintaining the efficient performance of the network.

scholarly journals Regional and seasonal truncation errors of trajectory calculations using ECMWF high-resolution operational analyses and forecasts

2017 ◽  
Author(s):  
Thomas Rößler ◽  
Olaf Stein ◽  
Yi Heng ◽  
Lars Hoffmann
Keyword(s):  
High Resolution ◽  
Numerical Integration ◽  
Transport Processes ◽  
Integration Scheme ◽  
Free Troposphere ◽  
Time Step ◽  
Truncation Errors ◽  
Trajectory Calculations ◽  
Integration Schemes ◽  
Runge Kutta

Abstract. Lagrangian particle dispersion models (LPDMs) are indispensable tools to study atmospheric transport processes. The accuracy of trajectory calculations, which form an essential part of LPDM simulations, depends on various factors. Here we focus on truncation errors that originate from the use of numerical integration schemes to solve the kinematic equation of motion. The optimization of numerical integration schemes to minimize truncation errors and to maximize computational speed is of great interest regarding the computational efficiency of large-scale LPDM simulations. In this study we analyzed truncation errors of six explicit integration schemes of the Runge Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) model. The simulations were driven by wind fields of the latest operational analysis and forecasts of the European Centre for Medium-range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct domains of the atmosphere, covering the polar regions, the mid-latitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the lower and mid stratosphere. For each domain we performed simulations for the months of January, April, July, and October for the years of 2014 and 2015. In total more than 5000 different transport simulations were performed. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a 4th-order Runge-Kutta integration scheme with a sufficiently fine time step. We assessed the transport deviations with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the truncation errors vary significantly between the different domains and seasons. Especially the differences in altitude stand out. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to generalize, we recommend the 3rd-order Runge Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days time based on ECMWF's high-resolution meteorological data.

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