Runge-Kutta methods for parabolic equations and convolution quadrature

Lubich, Ch.; Ostermann, A.

doi:10.1090/S0025-5718-1993-1153166-7

Runge-Kutta methods for parabolic equations and convolution quadrature
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by Ch. Lubich and A. Ostermann PDF

Math. Comp. 60 (1993), 105-131 Request permission

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 noninteger, 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.

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