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Mathematics of Computation

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Runge-Kutta methods for parabolic equations and convolution quadrature

Authors: Ch. Lubich and A. Ostermann
Journal: Math. Comp. 60 (1993), 105-131
MSC: Primary 65M12; Secondary 65D30, 65M15, 65R20, 76D05, 76M25
MathSciNet review: 1153166
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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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Keywords: Parabolic equations, nonstationary Navier-Stokes equation, Runge-Kutta time discretization, convolution integrals, numerical quadrature
Article copyright: © Copyright 1993 American Mathematical Society