Truncation error calculation based on Richardson extrapolation for variable-step collaborative simulation

Zhang, Heming; Liang, Silü; Song, Sejun; Wang, Hongwei

doi:10.1007/s11432-011-4274-z

Cited by 7 publications

(11 citation statements)

References 18 publications

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“…The multistep numerical methods for solving an ODE suffer from two distinct sources of error. Compared to truncation error, round-off error generally plays only a minor role [19,20]. Therefore, truncation error is the main focus of this research and the estimation of the p-order truncation error can be done using the following formula:…”

Section: Optimization Of the Accuracy Of Multistep Numerical Methods mentioning

confidence: 99%

An Accurate Method for Real-Time Aircraft Dynamics Simulation Based on Predictor-Corrector Scheme

Zhao

Wang

Zhang

2015

Mathematical Problems in Engineering

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Real-time aircraft dynamics simulation requires very high accuracy and stability in the numerical integration process. Nonetheless, traditional multistep numerical methods cannot effectively meet the new requirements. Therefore, a novel real-time multistep method based on Predict-Evaluate-Correct scheme of three-step fourth-order method (RTPEC-34) is proposed and developed in this research to address the gap. In addition to the development of a highly accurate algorithm based on predictor-corrector, the contribution of this work also includes the analysis of truncation error for real-time problems. Moreover, the parameters for the RTPEC-34 method are optimized using intelligent optimization algorithms. The application and comparison of the optimization algorithms also lead to general guidelines for their applications in the development of improved multistep methods. Last but not least, theoretical analysis is also conducted on the stability of the proposed RTPEC-34 method, which is corroborated in simulation experiments and thus provides general guidelines for the evaluation of real-time numerical methods. The RTPEC-34 method is compared with other multistep algorithms using both numerical experiments and a real engineering example. As shown in the comparison, it achieves improved performance in terms of accuracy and stability and it is also a viable and efficient algorithm for real-time aircraft dynamics simulation.

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Section: Optimization Of the Accuracy Of Multistep Numerical Methods mentioning

confidence: 99%

An Accurate Method for Real-Time Aircraft Dynamics Simulation Based on Predictor-Corrector Scheme

Zhao

Wang

Zhang

2015

Mathematical Problems in Engineering

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“…[6]) or by adaption before every macro step execution by predictive error estimation, f.i. via extrapolation or energy residuals [7,8]. Some loose coupling methods do not necessarily require any synchronized time steps from the sub-simulations apart from the start time, see f.i.…”

Section: Distinction By Macro Stepsmentioning

confidence: 99%

Analysis of Selected Literature on Co-simulation

Hafner¹,

Emrich²,

Popper³

2022

ASIM SST 2022 Proceedings Langbeiträge

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“…For example, the tool for control algorithm design may involve modeling using block diagrams while a multi-body dynamics package often has 3D modeling capability. The modeling of a MCS system [41] and the estimation of truncation errors [42] have been published elsewhere while this paper is focused on the development of a variable-step algorithm for the distributed and collaborative simulation of MCS systems.…”

Section: Coupling Between Models In Distributed Mcsmentioning

confidence: 99%

“…Then, the errors caused by the integration and interpolation methods can be shown as the left-hand sides of Eqs (10) and (11), respectively. Readers interested in the detailed derivation process of the formulas are referred to [42].…”

Section: Lte For Mcs Problemsmentioning

confidence: 99%

“…(12) In this work, the application of the Richardson extrapolation is extended from a single system to a coupled system such as MCS to derive the formulas for calculating the errors shown in Eqs (10) through to (12). The detailed derivation process of the formulas is out of the scope of this paper and has been published elsewhere [42]. Likewise, the calculation of LTEs for MCS problems also involves two steps of obtaining the results at step H/2 and H, respectively.…”

Section: Lte For Mcs Problemsmentioning

confidence: 99%

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A variable-step interaction algorithm for multidisciplinary collaborative simulation

Wang

Mao

Zhang

2014

ICA

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Multidisciplinary collaborative simulation is an important technique for the development of complex engineering systems such as aircrafts and automobiles. It emphasizes the synergic collaboration of multidisciplinary computational models, involving the exchange of simulation data generated in parallel at runtime from the numerical integration processes of these models. Current research on the interaction of models in multidisciplinary collaborative simulation is mainly focused on performing data exchange at fixed intervals, i.e. macro time steps, and using interpolation/extrapolation to find out values at the points in time when data are not available. However, the selection of an appropriate size for macro time tends to be complex and iterative. Firstly, it is mostly based on experience rather than the estimation of simulation errors by analyzing the numerical integration processes involved. Moreover, it is hardly possible to select a fixed step size suitable for all the stages of the simulation process. This paper presents a novel variable-step algorithm which is able to adjust the step size automatically at simulation runtime based on the evaluation of truncation errors. This algorithm not only avoids the instability and inaccuracy caused by a fixed large step but also speeds up the simulation by using a smaller step when a large one is not necessary. It is demonstrated in the numerical experiments that the variable-step algorithm can achieve improved simulation performance in terms of both speed and accuracy.

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Truncation error calculation based on Richardson extrapolation for variable-step collaborative simulation

Cited by 7 publications

References 18 publications

An Accurate Method for Real-Time Aircraft Dynamics Simulation Based on Predictor-Corrector Scheme

An Accurate Method for Real-Time Aircraft Dynamics Simulation Based on Predictor-Corrector Scheme

Analysis of Selected Literature on Co-simulation

A variable-step interaction algorithm for multidisciplinary collaborative simulation

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