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Analysis and Application of Multiple-Precision Computation and Round-off Error for Nonlinear Dynamical Systems


doi: 10.1007/s00376-006-0758-y

  • This research reveals the dependency of floating point computation in nonlinear dynamical systems on machine precision and step-size by applying a multiple-precision approach in the Lorenz nonlinear equations. The paper also demonstrates the procedures for obtaining a real numerical solution in the Lorenz system with long-time integration and a new multiple-precision-based approach used to identify the maximum effective computation time (MECT) and optimal step-size (OS). In addition, the authors introduce how to analyze round-off error in a long-time integration in some typical cases of nonlinear systems and present its approximate estimate expression.
  • [1] HU Shujuan, CHOU Jifan, 2004: Uncertainty of the Numerical Solution of a Nonlinear System's Long-term Behavior and Global Convergence of the Numerical Pattern, ADVANCES IN ATMOSPHERIC SCIENCES, 21, 767-774.  doi: 10.1007/BF02916373
    [2] Hongqin ZHANG, Xiangjun TIAN, Wei CHENG, Lipeng JIANG, 2020: System of Multigrid Nonlinear Least-squares Four-dimensional Variational Data Assimilation for Numerical Weather Prediction (SNAP): System Formulation and Preliminary Evaluation, ADVANCES IN ATMOSPHERIC SCIENCES, 37, 1267-1284.  doi: 10.1007/s00376-020-9252-1
    [3] XUE Hai-Le, SHEN Xue-Shun, CHOU Ji-Fan, 2013: A Forecast Error Correction Method in Numerical Weather Prediction by Using Recent Multiple-time Evolution Data, ADVANCES IN ATMOSPHERIC SCIENCES, 30, 1249-1259.  doi: 10.1007/s00376-013-2274-1
    [4] REN Hongli, CHOU Jifan, HUANG Jianping, ZHANG Peiqun, 2009: Theoretical Basis and Application of an Analogue- Dynamical Model in the Lorenz System, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 67-77.  doi: 10.1007/s00376-009-0067-3
    [5] Yang HE, Xiaoqian ZHU, Zheng SHENG, Wei GE, Xiaoran ZHAO, Mingyuan HE, 2022: Atmospheric Disturbance Characteristics in the Lower-middle Stratosphere Inferred from Observations by the Round-Trip Intelligent Sounding System (RTISS) in China, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 131-144.  doi: 10.1007/s00376-021-1110-2
    [6] Li Tianming, Zhu Yongti, 1989: On the Multiple Equilibrium of the Development of Tropical Cyclone in Nonlinear CISK Model, ADVANCES IN ATMOSPHERIC SCIENCES, 6, 447-456.  doi: 10.1007/BF02659078
    [7] SUN Guodong, MU Mu, 2009: Nonlinear Feature of the Abrupt Transitions between Multiple Equilibria States of an Ecosystem Model, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 293-304.  doi: 10.1007/s00376-009-0293-8
    [8] Zhao Jingxia, Zhu Baozhen, 1989: Sensitivity of the Multiple Equilibria to Gorverning System, Mode Chosen and Parameter, ADVANCES IN ATMOSPHERIC SCIENCES, 6, 33-43.  doi: 10.1007/BF02656916
    [9] Zhang Ren, Yu Zhihao, 2000: Numerical and Dynamical Analyses of Heat Source Forcing and Restricting Subtropical High Activity, ADVANCES IN ATMOSPHERIC SCIENCES, 17, 61-71.  doi: 10.1007/s00376-000-0043-4
    [10] Guo Yufu, Chao Jiping, 1984: SIMPLIFIED DYNAMICAL ANOMALY MODEL FOR LONG-RANGE NUMERICAL FORECASTS, ADVANCES IN ATMOSPHERIC SCIENCES, 1, 30-52.  doi: 10.1007/BF03187614
    [11] Si Gongwang, Donald R. Johnson, 1985: A NUMERICAL EXPERIMENTAL STUDY OF SOME DYNAMICAL EFFECTS ON SOUTHERN ASIATIC HIGH, ADVANCES IN ATMOSPHERIC SCIENCES, 2, 133-146.  doi: 10.1007/BF03179746
    [12] HUANG Sixun, CAO Xiaoqun, DU Huadong, WANG Tingfang, XIANG Jie, 2006: Retrieval of Atmospheric and Oceanic Parameters and the Relevant Numerical Calculation, ADVANCES IN ATMOSPHERIC SCIENCES, 23, 106-117.  doi: 10.1007/s00376-006-0011-8
    [13] Ruiqiang DING, Jianping LI, Baosheng LI, 2017: Determining the Spectrum of the Nonlinear Local Lyapunov Exponents in a Multidimensional Chaotic System, ADVANCES IN ATMOSPHERIC SCIENCES, 34, 1027-1034.  doi: 10.1007/s00376-017-7011-8
    [14] Zhizhen XU, Jing CHEN, Mu MU, Guokun DAI, Yanan MA, 2022: A Nonlinear Representation of Model Uncertainty in a Convective-Scale Ensemble Prediction System, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 1432-1450.  doi: 10.1007/s00376-022-1341-x
    [15] Ben TIAN, Hong-Li REN, 2022: Diagnosing SST Error Growth during ENSO Developing Phase in the BCC_CSM1.1(m) Prediction System, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 427-442.  doi: 10.1007/s00376-021-1189-5
    [16] LIU Juanjuan, WANG Bin, WANG Shudong, 2010: The Structure of Background-error Covariance in a Four-dimensional Variational Data Assimilation System: Single-point Experiment, ADVANCES IN ATMOSPHERIC SCIENCES, 27, 1303-1310.  doi: 10.1007/s00376-010-9067-6
    [17] XIA Zhiye, CHEN Hongbin, XU Lisheng, WANG Yongqian, 2015: Extended Range (10-30 Days) Heavy Rain Forecasting Study Based on a Nonlinear Cross-Prediction Error Model, ADVANCES IN ATMOSPHERIC SCIENCES, 32, 1583-1591.  doi: 10.1007/s00376-015-4252-2
    [18] Zhong Qing, Ji Liren, 1992: A Further Study on an Extended Nonlinear Ocean-Atmosphere Coupled Hydrodynamic Characteristic System and the Abrupt Feature of ENSO Events, ADVANCES IN ATMOSPHERIC SCIENCES, 9, 131-146.  doi: 10.1007/BF02657504
    [19] Ji-Hyun HA, Dong-Kyou LEE, 2012: Effect of Length Scale Tuning of Background Error in WRF-3DVAR System on Assimilation of High-Resolution Surface Data for Heavy Rainfall Simulation, ADVANCES IN ATMOSPHERIC SCIENCES, 29, 1142-1158.  doi: 10.1007/s00376-012-1183-z
    [20] Guo Weidong, Sun Shufen, Qian Yongfu, 2002: Case Analyses and Numerical Simulation of Soil Thermal Impacts on Land Surface Energy Budget Based on an Off-Line Land Surface Model, ADVANCES IN ATMOSPHERIC SCIENCES, 19, 500-512.  doi: 10.1007/s00376-002-0082-0

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Manuscript History

Manuscript received: 10 September 2006
Manuscript revised: 10 September 2006
通讯作者: 陈斌, bchen63@163.com
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Analysis and Application of Multiple-Precision Computation and Round-off Error for Nonlinear Dynamical Systems

  • 1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029,State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029,National Climate Center, Beijing 100081

Abstract: This research reveals the dependency of floating point computation in nonlinear dynamical systems on machine precision and step-size by applying a multiple-precision approach in the Lorenz nonlinear equations. The paper also demonstrates the procedures for obtaining a real numerical solution in the Lorenz system with long-time integration and a new multiple-precision-based approach used to identify the maximum effective computation time (MECT) and optimal step-size (OS). In addition, the authors introduce how to analyze round-off error in a long-time integration in some typical cases of nonlinear systems and present its approximate estimate expression.

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