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Runge-Kutta methods adapted to the numerical integration of oscillatory problems. (English) Zbl 1057.65043

The material is devoted to the construction of Runge-Kutta methods specially adapted to the numerical integration of initial value problems with oscillatory solutions. The coefficients of these methods are frequency-dependent such that certain particular oscillatory solutions are computed exactly.
The first section is given by an introduction concerning this type of Runge-Kutta methods.
Section two presents a detailed exposition of the ideas that lead to the formulation of the adapted Runge-Kutta methods. Explicit Runge-Kutta methods for oscillatory ordinary differential equations (ODEs) and order conditions are exposed. This part contains a modification of the classical Runge-Kutta algorithms such that the oscillatory linear problem is solved without truncation errors. Necessary and sufficient order conditions for the adapted Runge-Kutta methods are derived by using the B-series theory and the rooted trees. These conditions reduce to the classical order conditions for the classical Runge-Kutta methods when the parameter \(v= \omega h\to 0\), \(\omega\) being the main frequency of the oscillatory solutions and \(h\) the step-size.
In section three one derives explicit adapted Runge-Kutta methods with order 3 and 4 as well as embedded pairs of adapted Runge-Kutta methods of orders 3 and 4.
The fourth section presents some numerical experiments which support the efficiency of the adapted Runge-Kutta methods derived in the paper, when they are compared with other classical Runge-Kutta methods.
Two types of comparisons with both fixed and variable step-size are exposed, each one for a linear model problem test, respectively a nonlinear model problem test. Thus the numerical results show the excellent behaviour of the new methods when they are compared with standard Runge-Kutta methods.
Section five contains the conclusions and also some research perspective, like the extension of these methods to the case of semilinear ODE systems \((y'= Ay+ g(t, y))\), where the stiffness matrix \(A\) contains implicitly the frequencies of the problem.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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