Runge-Kutta methods adapted to the numerical integration of oscillatory problems

Authors:
J. M. Franco
Affiliations:
Departamento de Matemática Aplicada, C.P.S. de Ingeniería, Universidad de Zaragoza, María de Luna 3, 50015 Zaragoza, Spain
Venue:
Applied Numerical Mathematics
Year:
2004

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Abstract

New Runge-Kutta methods specially adapted to the numerical integration of IVPs with oscillatory solutions are obtained. The coefficients of these methods are frequency-dependent such that certain particular oscillatory solutions are computed exactly (without truncation errors). Based on the B-series theory and on the rooted trees we derive the necessary and sufficient order conditions for this class of RK methods. With the help of these order conditions we construct explicit methods (up to order 4) as well as pairs of embedded RK methods of orders 4 and 3. Some numerical examples show the excellent behaviour when they compete with classical RK methods.