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

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.