On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

2017 ◽  
Vol 8 (1-2) ◽  
pp. 139
Author(s):  
Truong Nguyen-Ba ◽  
Thierry Giordano ◽  
Huong Nguyen-Thu ◽  
Remi Vaillancourt
Keyword(s):  
Relative Energy ◽  
Positional Error ◽  
Maximum Order ◽  
Energy Error ◽  
Good Shape ◽  
Regions Of Stability ◽  
Nonnegative Coefficients ◽  
Contractivity Preserving ◽  
Predictor Corrector ◽  
Numer Math

The contractivity-preserving 2- and 3-step predictor-corrector series methods for ODEs  (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt,  On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs,  J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17--39. doi:10.20454/jmmnm.2017.1130)  are expanded into  new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector,  Hermite- Birkhoff--Obrechkoff series methods, denoted by HBO(d,k,p), k=4,5,6,7, with nonnegative coefficients  for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\).  The main reason for considering this class of formulae is to obtain a set of methods  which have larger regions of stability and generally higher upper bound \(p_u\) of  order \(p\) of HBO(d,k,p) for a given d. Their stability regions have generally  a good shape and grow generally with decreasing \(p-d\).  A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14)  which has maximum order 14 based on the CP conditions compares satisfactorily  with Adams--Cowell of order 13 in PECE mode, denoted by AC(13),  in solving standard N-body problems over an interval of 1000 periods  on the basis of the relative error of energy as a function of the CPU time.  HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems  on the basis of the growth of relative positional error, relative energy error  and 10000 periods of integration.  The coefficients of HBO(6,4,14) are listed in the appendix.

On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs

2017 ◽  
Vol 8 (1-2) ◽  
pp. 17
Author(s):  
Truong Nguyen-Ba ◽  
Abdulrahman Alzahrani ◽  
Thierry Giordano ◽  
Remi Vaillancourt
Keyword(s):  
Relative Error ◽  
Upper Bounds ◽  
Initial Value ◽  
Maximum Order ◽  
First Order ◽  
Cpu Time ◽  
Good Shape ◽  
Nonnegative Coefficients ◽  
Contractivity Preserving ◽  
Predictor Corrector

New optimal, contractivity-preserving (CP), \(d\)-derivative, 2- and 3-step, predictor-corrector,  Hermite-Birkhoff-Obrechkoff series methods, denoted by \(HBO(d,k,p)\), \(k=2,3\), with  nonnegative coefficients are constructed  for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\).  The upper bounds \(p_u\) of order \(p\) of \(HBO(d,k,p)\), \(k=2,3\) methods are approximately 1.4 and 1.6 times the number  of derivatives \(d\), respectively.  Their stability regions have generally  a good shape and grow with decreasing \(p-d\).  Two selected CP HBO methods: 9-derivative 2-step HBO of order 13, denoted by HBO(9,2,13),  which has maximum order 13 based on the CP conditions, and  8-derivative 3-step HBO of order 14, denoted by HBO(8,3,14), compare well  with Adams-Cowell of order 13 in PECE mode, denoted by AC(13),  in solving standard N-body problems over an interval of 1000 periods  on the basis of the relative error of energy as a function of the CPU time.  They also compare well with AC(13) in solving standard N-body problems on the basis of the growth of relative error of energy and 10000 periods of integration.  The coefficients of selected HBO methods are listed in the  appendix.

Modeling the dynamic behavior of laminated steels using a Fourier-based approach

2020 ◽  
Vol 92 (1) ◽  
pp. 10905
Author(s):  
Reza Zeinali ◽  
Dave Krop ◽  
Elena Lomonova
Keyword(s):  
Fourier Series ◽  
Dynamic Model ◽  
Eddy Current ◽  
Relative Energy ◽  
System Of Nonlinear Equations ◽  
Hysteresis Phenomenon ◽  
Hysteresis Model ◽  
Energy Error ◽  
Dynamic Hysteresis ◽  
Lamination Thickness

A new magneto-dynamic model is proposed to approximate the dynamic hysteresis effect in laminated steels considering the static hysteresis, eddy-current field, and excess field. An accurate congruency-based hysteresis model is used to predict the static hysteresis field. The eddy-current is determined from the 1D diffusion equation and the well-known Bertotti empirical equation is utilized to model the excess-field effect. The dynamic lamination model obtained from coupling three field components is solved using a Fourier-based approach. In this approach, the flux density across the lamination thickness is approximated by a cosine-based Fourier series. The coefficients of the Fourier series are determined by solving a system of nonlinear equations through an iterative procedure. Owing to the employed congruency-based static hysteresis model, the proposed magneto-dynamic model offers high accuracy for arbitrary magnetization regimes. To validate the model accuracy, the model results are compared with sinusoidal and multi-harmonic measurements. The comparison shows that the proposed model predicts the dynamic hysteresis phenomenon in laminated steels with a relative energy error of less than 7%.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

The Possibilities for Analytical Methods in Photoemission and Low-Energy Electron Microscopy

1990 ◽  
Vol 48 (1) ◽  
pp. 348-349
Author(s):  
Ernst Bauer
Keyword(s):  
Chemical Characterization ◽  
Emission Angle ◽  
Relative Energy ◽  
Objective Lens ◽  
Chemical Information ◽  
Magnetic Lens ◽  
Incident Intensity ◽  
X Ray ◽  
Leed Pattern ◽  
Image Position

One of the major shortcomings of conventional PEEM and of LEEM is the lack of chemical information about the surface. Although the imaging of the LEED pattern in the back focal plane of the objective lens of a LEEM instrument allows chemical characterization via the crystalline structure derived from the LEED pattern, this method fails in the absence of a characteristic LEED pattern. Direct information about the atomic composition of the surface is then needed which can be best obtained from inner shell electrons either directly by x-ray-induced photoemission (XPEEM) or by x-ray- or electron-induced Auger electron emission (AEEM). These modes of excitation and imaging can be combined with conventional PEEM and LEEM in one instrument which is presently being developed. Thus a complete structural and chemical characterization becomes possible in one instrument, with parallel detection and high resolution.In contrast to LEEM, in which up to more than 50% of the incident intensity is available for image formation, the intensity of the emitted electrons is much lower in XPEEM and AEEM and the signal is much lower than the background in AEEM. Therefore, intensity I and resolution d have to be optimized simultaneously which is best done by maximizing Q = I/d2 with respect to maximum emission angle α and relative energy distribution ε = ΔVo/V accepted by the instrument. For a well-designed magnetic lens section of the cathode lens its aberrations are determined by the accelerating field F in front of the specimen. For a homogeneous accelerating field F and a cosine emission distribution one obtains for the optimum α and ε values αo,εo a radius of the minimum disc of confusion of

THE INFLUENCE OF THE WIND SPEED PREDICTION ERROR ON THE SIZE OF THE STORAGE CONTROLLED OPERATION ZONE IN THE SYSTEM WITH THE WIND GENERATOR

2020 ◽  
Vol 2020 (57) ◽  
pp. 35-41
Author(s):  
K.S. Klen ◽  
◽  
M.K. Yaremenko ◽  
V.Ya. Zhuykov ◽  
◽  
...  
Keyword(s):  
Wind Speed ◽  
Prediction Error ◽  
Average Energy ◽  
Interpolation Formula ◽  
Wind Speed Prediction ◽  
Wind Generator ◽  
Prediction Horizon ◽  
Speed Prediction ◽  
Using Data ◽  
Predictor Corrector

The article analyzes the influence of wind speed prediction error on the size of the controlled operation zone of the storage. The equation for calculating the power at the output of the wind generator according to the known values of wind speed is given. It is shown that when the wind speed prediction error reaches a value of 20%, the controlled operation zone of the storage disappears. The necessity of comparing prediction methods with different data discreteness to ensure the minimum possible prediction error and determining the influence of data discreteness on the error is substantiated. The equations of the "predictor-corrector" scheme for the Adams, Heming, and Milne methods are given. Newton's second interpolation formula for interpolation/extrapolation is given at the end of the data table. The average relative error of MARE was used to assess the accuracy of the prediction. It is shown that the prediction error is smaller when using data with less discreteness. It is shown that when using the Adams method with a prediction horizon of up to 30 min, within ± 34% of the average energy value, the drive can be controlled or discharged in a controlled manner. References 13, figures 2, tables 3.

Issues and Challenges of Asian Development

Worldview ◽  
1975 ◽  
Vol 18 (7-8) ◽  
pp. 36-39
Author(s):  
Saburo Okita
Keyword(s):  
Growth Opportunities ◽  
Asian Development ◽  
Good Shape ◽  
Monetary System ◽  
The Social ◽  
The World ◽  
Short And Long Term ◽  
Oil Crisis ◽  
Complicated Nature

The economy of Southeast Asia has been in relatively good shape in spite of the instability of the world monetary system, trade deficits, and the worldwide oil crisis. There are promising factors for economic growth, opportunities for employment, and possibilities of rising income. But Asian development presents short-and long-term problems of a very complicated nature. One of the most serious problems is inflation and its impact on the social and political programs of individual countries. At the same time, there are severe shortages of basic commodities, such as oil and food. My own country, Japan, is among those affected.

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