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Runge-Kutta Methods for parabolic equations and convolution quadrature

by Ch. Lubich and A. Ostermann

(Report number 1991-06)

Abstract
We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally non-integer, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the convolution kernel via a discrete operational calculus.

Keywords: Parabolic equations, nonstationary Navier-Stokes equation, Runge-Kutta time discretization, convolution integrals, numerical quadrature

@Techreport{LO91_6,
  author = {Ch. Lubich and A. Ostermann},
  title = {Runge-Kutta Methods for parabolic equations and convolution quadrature},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1991-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1991/1991-06.pdf },
  year = {1991}
}

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