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Runge-Kutta solutions of stiff differential

by Ch. Lubich and K. Nipp and D. Stoffer

(Report number 1993-01)

Abstract
Runge-Kutta methods applied to stiff systems in singular perturbation form are shown to give accurate approximations of phase portraits near hyperbolic stationary points. We prove that Runge-Kutta solutions shadow solutions of the differential equation over arbitrarily long time intervals, and vice versa. Sharp error estimates are derived. The proof uses attractive invariant manifolds to reduce the problem to the nonstiff case which was previously studied by Beyn.

Keywords: Long-time error bounds, implicit Runge-Kutta method, stiff ODE, singular perturbation problem, hyperbolic equilibrium, shadowing, invariant manifold.

@Techreport{LNS93_26,
  author = {Ch. Lubich and K. Nipp and D. Stoffer},
  title = {Runge-Kutta solutions of stiff differential},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1993-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1993/1993-01.pdf },
  year = {1993}
}

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