An efficient method for solving implicit and explicit stiff differential equations.

*(English)*Zbl 0962.65057This paper deals with an efficient algorithm which applies to both explicit as well as implicit ordinary differential equations. It differs from the traditional Runge-Kutta method. Using a polynomial of degree \(s\) for the solution \(y(x)\) leads to a nonlinear system which is solved by the Newton method. The formal precedure for uncoupling the algebraic system into a block-diagonal matrix with \(s\) blocks of size \(n\) is derived for any virtual number of stages \(s\).

The method may easily be constructed to be either \(A\)- or \(L\)-stable. In particular, for \(s= 3\) it has the same precision and stability properties as the well-known Runge-Kutta based Radau IIA method. Further, it may be considered as a good candidate for solving differential-algebraic equations of even higher index. Finally, the method is demonstrated by its application in the multibody dynamics.

The method may easily be constructed to be either \(A\)- or \(L\)-stable. In particular, for \(s= 3\) it has the same precision and stability properties as the well-known Runge-Kutta based Radau IIA method. Further, it may be considered as a good candidate for solving differential-algebraic equations of even higher index. Finally, the method is demonstrated by its application in the multibody dynamics.

Reviewer: I.Čomić (Novi Sad)

##### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

65L80 | Numerical methods for differential-algebraic equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

##### Keywords:

initial value problem; multibody dynamics; numerical examples; stiff systems; comparison of methods; Newton method; stability; differential-algebraic equations##### Software:

PSIDE
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\textit{H. Pasic}, Int. J. Numer. Methods Eng. 48, No. 1, 55--78 (2000; Zbl 0962.65057)

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