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)
Trigonometrically fitted explicit Numerov-type method for periodic ivps with two frequencies. (English) Zbl 1197.65085
Summary: New trigonometrically fitted Numerov type methods for the periodic initial problems are proposed. These methods are based on the original Numerov-type sixth order method with fifth internal stages motivated by Ch. Tsitouras [Comput. Math. Appl. 45, 37–42 (2003; Zbl 1035.65078)]. Some parameters are added to these methods so that they can integrate exactly the combination of trigonometrically functions with two frequencies. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.
MSC:
65L06Multistep, Runge-Kutta, and extrapolation methods
65L05Initial value problems for ODE (numerical methods)
References:
[1]Coleman, J. P.: Order conditions for a class of two-step methods for y″=f(x,y), IMA J. Numer. anal. 23, 197-220 (2003) · Zbl 1022.65080 · doi:10.1093/imanum/23.2.197
[2]Coleman, J. P.; Ixaru, L. Gr.: P-stability and exponential-Fitting methods for y″=f(x,y), IMA J. Numer. anal. 16, 179-199 (1996)
[3]Fang, Y. L.; Wu, X. Y.: A trigonometrically fitted explicit numerov-type method for second-order initial value problems with oscillating solutions, Appl. numer. Math. 58, 341-351 (2008) · Zbl 1136.65068 · doi:10.1016/j.apnum.2006.12.003
[4]Franco, J. M.: A class of explicit two-step hybrid methods for second-order ivps, J. comput. Appl. math. 187, 41-57 (2006) · Zbl 1082.65071 · doi:10.1016/j.cam.2005.03.035
[5]Franco, J. M.: Exponentially fitted explicit Runge – Kutta – Nyström methods, J. comput. Appl. math. 167, 1-19 (2004) · Zbl 1060.65073 · doi:10.1016/j.cam.2003.09.042
[6]Franco, J. M.: Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order, Numer. algor. 26, 347-363 (2001) · Zbl 0974.65076 · doi:10.1023/A:1016629706668
[7]Kramarz, L.: Stability of collocation methods for the numerical solution of y″=f(t,y), Bit 20, 215-222 (1980) · Zbl 0425.65043 · doi:10.1007/BF01933194
[8]Lambert, J. D.; Watson, I. A.: Symmetric multistep methods for periodic initial value problems, J. inst. Math. appl. 18, 189-202 (1976) · Zbl 0359.65060 · doi:10.1093/imamat/18.2.189
[9]Tsitouras, Ch.: Explicit numerov type methods with reduced number of stages, Comput. math. Appl. 45, 37-42 (2003) · Zbl 1035.65078 · doi:10.1016/S0898-1221(03)80005-6
[10]Van Der Houwen, P. J.; Sommeijer, B. P.: Explicit Runge – Kutta(--Nyström) methods with reduced phase errors for computing oscillating solution, SIAM J. Numer. anal. 24, 595-617 (1987) · Zbl 0624.65058 · doi:10.1137/0724041
[11]Berghe, G. Vanden; De Meyer, H.; Van Daele, M.; Van Hecke, T.: Exponentially fitted Runge – Kutta methods, Comput. phys. Comm. 123, 7-15 (1999) · Zbl 0948.65066 · doi:10.1016/S0010-4655(99)00365-3
[12]Berghe, G. Vanden; De Meyer, H.; Van Daele, M.; Van Hecke, T.: Exponentially fitted explicit Runge – Kutta methods, J. comput. Appl. math. 125, 107-115 (2000) · Zbl 0999.65065 · doi:10.1016/S0377-0427(00)00462-3
[13]Van De Vyver, H.: Stability and phase-lag analysis of explicit Runge – Kutta methods with variable coefficients for oscillatory problems, Comput. phys. Comm. 173, 115-130 (2005) · Zbl 1196.65117 · doi:10.1016/j.cpc.2005.07.007
[14]Van De Vyver, H.: An explicit numerov-type method for second-order differential equations with oscillating solutions, Comput. math. Appl. 53, 1339-1348 (2007) · Zbl 1121.65086 · doi:10.1016/j.camwa.2006.06.012
[15]Wang, Z. C.: 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