zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the frequency choice in trigonometrically fitted methods. (English) Zbl 1197.65082
Summary: The choice of frequency in trigonometrically fitted methods is a fundamental question, especially if long-term prediction is considered. For linear oscillators, the frequency of the method is the same as the frequency of the solution of the differential equation. However, for nonlinear problems the frequency of the method is, in general, different from the frequency of the true solution. We present some experiments showing how the frequency depends strongly on certain values.
MSC:
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
References:
[1]Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. math. 3, 381-397 (1961) · Zbl 0163.39002 · doi:10.1007/BF01386037
[2]Vigo-Aguiar, J.; Ferrándiz, J. M.: A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems, SIAM J. Numer. anal. 35, 1684-1708 (1998) · Zbl 0916.65081 · doi:10.1137/S0036142995286763
[3]Simos, T. E.; Vigo-Aguiar, J.: An exponentially-fitted high order method for long-term integration of periodic initial-value problems, Comput. phys. Comm. 140, 358-365 (2001) · Zbl 0991.65063 · doi:10.1016/S0010-4655(01)00285-5
[4]Franco, J. M.: Runge–Kutta methods adapted to the numerical integration of oscillatory problems, Appl. numer. Math. 50, 427-443 (2004) · Zbl 1057.65043 · doi:10.1016/j.apnum.2004.01.005
[5]Wang, Zhongcheng: Trigonometrically-fitted method for a periodic initial value problem with two frequencies, Comput. phys. Comm. 175, 241-249 (2006) · Zbl 1196.65137 · doi:10.1016/j.cpc.2006.03.004
[6]Fang, Yonglei; Wu, Xinyuan: A trigonometrically fitted explicit numerov-type method for second-order initial value problems with oscillating solutions, Appl. numer. Math. 585, 341-351 (2008) · Zbl 1136.65068 · doi:10.1016/j.apnum.2006.12.003
[7]Ixaru, L. Gr.; Rizea, M.; De Meyer, H.; Berghe, G. Vanden: Weights of the exponential Fitting multistep algorithms for odes, J. comput. Appl. math. 132, 83-93 (2001) · Zbl 0991.65061 · doi:10.1016/S0377-0427(00)00599-9
[8]Ixaru, L. Gr.; Berghe, G. Vanden; De Meyer, H.: Frequency evaluation in exponential Fitting multistep algorithms for odes, J. comput. Appl. math. 140, 423-434 (2002) · Zbl 0996.65075 · doi:10.1016/S0377-0427(01)00474-5
[9]Berghe, G. Vanden; Ixaru, L. Gr.; Van Daele, M.: Optimal implicit exponentially-fitted Runge–Kutta methods, Comput. phys. Comm. 140, 346-357 (2001) · Zbl 0990.65080 · doi:10.1016/S0010-4655(01)00279-X
[10]Van Daele, M.; Berghe, G. Vanden: Geometric numerical integration by means of exponentially-fitted methods, Appl. numer. Math. 57, 415-435 (2007) · Zbl 1116.65125 · doi:10.1016/j.apnum.2006.06.001
[11]Berghe, G. Vanden; De Meyer, H.; Vanthournout, J.: A modified numerov integration method for second order periodic initial-value problems, Int. J. Comput. math. 32, 233-242 (1990) · Zbl 0752.65059 · doi:10.1080/00207169008803830
[12]Beléndez, A.; Beléndez, T.; Márquez, A.; Neipp, C.: Application of he’s homotopy perturbation method to conservative truly nonlinear oscillators, Chaos solitons fractals 37, 770-780 (2008) · Zbl 1142.65055 · doi:10.1016/j.chaos.2006.09.070