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Integration of stiff mechanical systems by Runge-Kutta methods

by Ch. Lubich

(Report number 1992-04)

Abstract
The numerical integration of stiff mechanical systems is studied in which a strong potential forces the motion to remain close to a manifold. The equations of motion are written as a singular singular perturbation problem with a small stiffness parameter epsilon. Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than epsilon are shown to approximate smooth solutions, and precise error estimates are derived. As epsilon -> 0, Runge-Kutta solutions of the stiff system converge to Runge-Kutta solutions of the associated constrained system formulated as a differential-algebraic equation of index 3. Standard software for stiff initial-value problems does not work satisfactorily on the stiff systems considered here. The reasons of this failure are explained, and remedies are proposed.

Keywords: stiff mechanical system, stiff ODE,singular singular perturbationproblem, differential-algebraic equations, Runge-Kutta methods

@Techreport{L92_14,
  author = {Ch. Lubich},
  title = {Integration of stiff mechanical systems by Runge-Kutta methods},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1992-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1992/1992-04.pdf },
  year = {1992}
}

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