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 pap...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 demoastrates 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.展开更多
The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components o...The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multistep methods are essentially improved, and joining probabilistic theory the “normal” growth of accumulated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numerical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Furthermore, we give the explicitly mathematical expression of the computational uncertainty principle and expound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.展开更多
In a majority of cases of long-time numerical integration for initial-value problems, roundoff error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical so...In a majority of cases of long-time numerical integration for initial-value problems, roundoff error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical solutions is generally studied through a large number of numerical experiments. Here we find that there exists a strong dependence on machine precision (which is a new kind of dependence different from the sensitive dependence on initial conditions), maximally effective computation time (MECT) and optimal stepsize (OS) in solving nonlinear ordinary differential equations (ODEs) in finite machine precision. And an optimal searching method for evaluating MECT and OS under finite machine precision is presented. The relationships between MECT, OS, the order of numerical method and machine precision are found. Numerical results show that round-off error plays a significant role in the above phenomena. Moreover, we find two universal relations which are independent of the types of ODEs, initial values and numerical schemes. Based on the results of numerical experiments, we present a computational uncertainty principle, which is a great challenge to the reliability of long-time numerical integration for nonlinear ODEs.展开更多
With the proposition of the Digital Earth(DE)concept,Virtual Geographic Information System(VGIS)has started to play the role of a Digital Earth prototype system.Many core problems involved in VGIS,such as out-of-core ...With the proposition of the Digital Earth(DE)concept,Virtual Geographic Information System(VGIS)has started to play the role of a Digital Earth prototype system.Many core problems involved in VGIS,such as out-of-core management and interactive rendering of very large scale terrain and image data,have been well studied in the past decades.However,the jitter problem,a common problem in VGIS that often causes annoying visual artefacts and deteriorates the output image quality,draws little attention.In this paper,after an intensive analysis of the jitter problem,a comprehensive framework is proposed to address such a problem while accounting for the characteristics of different data types in VGIS,such as terrain or ocean mesh data,vector data and 3-D model data.Specifically,this framework provides an improved dynamic local coordinate system(DLCS)method for terrain or ocean mesh data.For vector data,the framework provides a simple and effective multiple local coordinate systems(MLCS)method.The framework provides a MLCS method for 3-D model data making full use of the existing local coordinate system of the model.The advantages of the proposed methods over current approaches are analysed and highlighted through case studies involving large GIS datasets.展开更多
In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allow...In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.展开更多
基金This study was supported by the National Key Basic Research and Development Project of China 2004CB418303 the National Natural Science foundation of China under Grant Nos. 40305012 and 40475027Jiangsu Key Laboratory of Meteorological Disaster KLME0601.
文摘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 demoastrates 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.
基金This work was supported by the Knowledge Innovation Key Project of Chinese Academy of Sciences inthe Resource Environment Field (KZCX1-203) Outstanding State Key Laboratory Project (Grant No. 49823002) the National Natural Science Foundation of C
文摘The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multistep methods are essentially improved, and joining probabilistic theory the “normal” growth of accumulated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numerical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Furthermore, we give the explicitly mathematical expression of the computational uncertainty principle and expound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.
文摘In a majority of cases of long-time numerical integration for initial-value problems, roundoff error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical solutions is generally studied through a large number of numerical experiments. Here we find that there exists a strong dependence on machine precision (which is a new kind of dependence different from the sensitive dependence on initial conditions), maximally effective computation time (MECT) and optimal stepsize (OS) in solving nonlinear ordinary differential equations (ODEs) in finite machine precision. And an optimal searching method for evaluating MECT and OS under finite machine precision is presented. The relationships between MECT, OS, the order of numerical method and machine precision are found. Numerical results show that round-off error plays a significant role in the above phenomena. Moreover, we find two universal relations which are independent of the types of ODEs, initial values and numerical schemes. Based on the results of numerical experiments, we present a computational uncertainty principle, which is a great challenge to the reliability of long-time numerical integration for nonlinear ODEs.
基金by the National High Technology Research and Development Program of China(863 Program)(No.2009AA12Z331)and the National Natural Science Foundation of China(No.60972052)supported by the Young Researcher Grant of National Astronomical Observatories,Chinese Academy of Sciences.
文摘With the proposition of the Digital Earth(DE)concept,Virtual Geographic Information System(VGIS)has started to play the role of a Digital Earth prototype system.Many core problems involved in VGIS,such as out-of-core management and interactive rendering of very large scale terrain and image data,have been well studied in the past decades.However,the jitter problem,a common problem in VGIS that often causes annoying visual artefacts and deteriorates the output image quality,draws little attention.In this paper,after an intensive analysis of the jitter problem,a comprehensive framework is proposed to address such a problem while accounting for the characteristics of different data types in VGIS,such as terrain or ocean mesh data,vector data and 3-D model data.Specifically,this framework provides an improved dynamic local coordinate system(DLCS)method for terrain or ocean mesh data.For vector data,the framework provides a simple and effective multiple local coordinate systems(MLCS)method.The framework provides a MLCS method for 3-D model data making full use of the existing local coordinate system of the model.The advantages of the proposed methods over current approaches are analysed and highlighted through case studies involving large GIS datasets.
基金The work of both authors has been supported by the NSF under Grant No.DMS-0608844.
文摘In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.
