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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.

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.

Reviewer: R. Militaru (Craiova)

### 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 |

### Keywords:

adapted Runge-Kutta methods; oscillatory problems; B-series theory; step-size control; comparison of methods; oscillatory solutions; order conditions; rooted trees; numerical experiments
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\textit{J. M. Franco}, Appl. Numer. Math. 50, No. 3--4, 427--443 (2004; Zbl 1057.65043)

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