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

DOI10.1016/S0168-9274(96)00038-4zbMath0872.65090OpenAlexW2047020621MaRDI QIDQ5961747

Alexander Ostermann, Christian Lubich

Publication date: 8 October 1997

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/s0168-9274(96)00038-4

zbMATH Keywords

Navier-Stokes equationserror boundslong-time behaviorreaction-diffusion equationssemilinear parabolic equationsasymptotically stable periodic orbitsRunge-Kutta time discretization

Mathematics Subject Classification ID

Reaction-diffusion equations (35K57) Navier-Stokes equations (35Q30) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Error bounds for initial value and initial-boundary value problems involving PDEs (65M15) Method of lines for initial value and initial-boundary value problems involving PDEs (65M20)

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Uses Software

• RODAS

Cites Work

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• Inertial manifolds for partial differential evolution equations under time-discretization: Existence, convergence, and applications
• Geometric theory of semilinear parabolic equations
• On the discretization of a partial differential equation in the neighborhood of a periodic orbit
• On invariant closed curves for one-step methods
• On the Navier-Stokes initial value problem. I
• Semidiscretization in Time for Parabolic Problems
• Finite Element Approximation of the Nonstationary Navier–Stokes Problem, Part II: Stability of Solutions and Error Estimates Uniform in Time
• On the Discretization in Time of Semilinear Parabolic Equations with Nonsmooth Initial Data
• Upper Semicontinuity of Attractors for Approximations of Semigroups and Partial Differential Equations
• On the qualitative behavior of the orbits of a parabolic partial differential equation and its discretization in the neighborhood of a hyperbolic fixed point
• Runge-Kutta Methods for Parabolic Equations and Convolution Quadrature
• The Behavior of Finite Element Solutions of Semilinear Parabolic Problems Near Stationary Points