Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour

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Abstract

We derive error bounds for Runge-Kutta time discretizations of semilinear parabolic equations with nonsmooth initial data. The framework includes reaction-diffusion equations and the incompressible Navier-Stokes equations. Nonsmooth-data error bounds of the type given here are needed in the study of the long-time behaviour of numerical discretizations. As an illustration, we use these low-order error bounds in proving high-order convergence of invariant closed curves of a Runge-Kutta method to periodic orbits of the parabolic problem.

References (17)

  • F. Demengel et al.

    Inertial manifolds for partial differential evolution equations under time discretization: existence, convergence, and applications

    J. Math. Anal. Appl.

    (1991)
  • F. Alouges et al.

    On the qualitative behavior of the orbits of a parabolic partial differential equation and its discretization in the neighborhood of a hyperbolic fixed point

    Numer. Funct. Anal. Optim.

    (1991)
  • F. Alouges et al.

    On the discretization of a partial differential equation in the neighborhood of a periodic orbit

    Numer. Math.

    (1993)
  • W.-J. Beyn

    On invariant closed curves for one-step methods

    Numer. Math.

    (1987)
  • J.C. Butcher

    The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods

    (1987)
  • M. Crouzeix et al.

    On the discretization in time of semilinear parabolic equations with nonsmooth initial data

    Math. Comp.

    (1987)
  • H. Fujita et al.

    On the Navier-Stokes initial-value problem

    Arch. Rational Mech. Anal.

    (1964)
  • J.K. Hale et al.

    Upper semicontinuity of attractors for approximations of semigroups and partial differential equations

    Math. Comp.

    (1988)
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