论文信息 - Integration of stiff mechanical systems by Runge-Kutta methods

Integration of stiff mechanical systems by Runge-Kutta methods

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 ɛ. Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than ɛ are shown to approximate smooth solutions, and precise error estimates are derived. As ɛ → 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 for this failure are explained, and remedies are proposed.

C. Lubich

[1] D. Morgenstern,et al. Vorlesungen Über Theoretische Mechanik , 1961 .

[2] Tosio Kato. Perturbation theory for linear operators , 1966 .

[3] Ralph Abraham,et al. Foundations Of Mechanics , 2019 .

[4] John C. Butcher,et al. On the implementation of implicit Runge-Kutta methods , 1976 .

[5] Ernst Hairer,et al. Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations , 1988 .

[6] J. Rappaz,et al. On numerical approximation in bifurcation theory , 1990 .

[7] J. R. E. O’Malley. Singular perturbation methods for ordinary differential equations , 1991 .

[8] K. Nipp,et al. On the Application of Invariant Manifold Theory, in particular to Numerical Analysis , 1991 .

[9] L. Jay. Collocation methods for differential-algebraic equations of index 3 , 1993 .

[10] Robert E. O'Malley,et al. Regularization of nonlinear differential-algebraic equations , 1994 .

[11] Linda R. Petzold,et al. Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.