# 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)
Exponentially fitted Runge-Kutta methods. (English) Zbl 0999.65065
Summary: Exponentially fitted Runge-Kutta methods with $s$ stages are constructed, which exactly integrate differential initial-value problems whose solutions are linear combinations of functions of the form $\{x^j\exp(\omega x),x^j\exp(-\omega x)\}$ $(\omega\in{\Bbb R}$ or $i{\Bbb R}, j=0,1,\dots,j_{\max})$ where $0\leq j_{\max}\leq\lfloor s/2-1\rfloor$, the lower bound being related to explicit methods and the upper bound applicable for collocation methods. Explicit methods with $s\in\{2,3,4\}$ belonging to that class are constructed. For these methods, a study of the local truncation error is made, from which follows a simple heuristic to estimate the $\omega$-value. Error and step length control are introduced based on Richardson extrapolation ideas. Some numerical experiments show the efficiency of the methods introduced. It is shown that the same techniques can be applied to construct implicit exponentially fitted Runge-Kutta methods.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE
Full Text:
##### References:
 [1] Butcher, J. C.: The numerical analysis of ordinary differential equations. (1987) · Zbl 0616.65072 [2] Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. math. 3, 381-397 (1961) · Zbl 0163.39002 [3] Hairer, E.; Nørsett, S. P.; Wanner, G.: Solving ordinary differential equations I, nonstiff problems. (1993) · Zbl 0789.65048 [4] Ixaru, L.: Operations on oscillatory functions. Comput. phys. Comm. 105, 1-19 (1997) · Zbl 0930.65150 [5] Lyche, T.: Chebyshevian multistep methods for ordinary differential equations. Numer. math. 19, 65-75 (1972) · Zbl 0221.65123 [6] Paternoster, B.: Runge--$Kutta(--Nystr\"om)$ methods for odes with periodic solutions based in trigonometric polynomials. Appl. numer. Math. 28, 401-412 (1998) · Zbl 0927.65097 [7] Simos, T. E.: An exponentially fitted Runge--Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. phys. Comm. 115, 1-8 (1998) · Zbl 1001.65080 [8] Simos, T. E.; Dimas, E.; Sideridis, A. B.: A Runge--Kutta--Nyström method for the numerical integration of special second-order periodic initial-value problems. J. comput. Appl. math. 51, 317-326 (1994) · Zbl 0872.65066 [9] Van Dooren, R.: Stabilization of cowell’s classical finite difference methos for numerical integration. J. comput. Phys. 16, 186-192 (1974) · Zbl 0294.65042 [10] Berghe, G. Vanden; De Meyer, H.; Van Daele, M.; Van Hecke, T.: Exponentially fitted explicit Runge--Kutta methods. Comput. phys. Comm. 123, 7-15 (1999) · Zbl 0948.65066