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Generalized disks of contractivity for explicit and implicit Runge-Kutta methods

by G. Dahlquist and R. Jeltsch

(Report number 2008-20)

Abstract
The A-contractivity of Runge-Kutta methods with respect to an inner-product norm, was investigated thoroughly by Butcher and Burrage (who used the term B-stability). Their theory is here extended to contractivity in a circular region tangential to the imaginary axis at the origin. The largest possible circle is calculated for many known explicit Runge-Kutta methods. As a rule it is considerably smaller than the stability region, and in several cases it degenerates to a point.

Keywords:

@Techreport{DJ08_384,
  author = {G. Dahlquist and R. Jeltsch},
  title = {Generalized disks of contractivity for explicit and implicit Runge-Kutta methods},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-20.pdf },
  year = {2008}
}

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