Abstract
We study semilinear evolution equations frac{mathrm dU}{mathrm dt}=AU+B(U) posed on a Hilbert space mathcal Y, where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from mathcal Y_ell = D(A^ell ) to itself, and ell in I subseteq [0,L], L ge 0, 0,L in I. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge–Kutta method in time, retaining continuous space, and prove convergence of order O(h^{pell /(p+1)}) for non-smooth initial data U^0in mathcal Y_ell , where ell le p+1, for a method of classical order p, extending a result by Brenner and Thomée for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp.
Highlights
We study the convergence of a class of A-stable Runge–Kutta time semidiscretizations of the semilinear evolution equation dU = AU + B(U )
In the examples we have in mind (1.1) is a partial differential equation (PDE)
We assume that (1.1) is posed on a Hilbert space Y, A is a normal linear operator that generates a strongly continuous semigroup, and that B is smooth on a scale of Hilbert spaces {Y } ∈I, I ⊆ [0, L], 0, L ∈ I, as detailed in condition (B) below
Summary
Related results include those of Brenner and Thomée [3], who consider linear evolution equations U = AU in a more general setting, namely posed on a Banach space X , where A generates a strongly continuous semigroup et A on X They show O(hq ) convergence of A-acceptable rational approximations of the semigroup for non-smooth initial data U 0 ∈ D(A ), = 0, . In particular in [12, Theorems 4.1 and 4.2] the existence of ( ps + 2) time derivatives of the continuous solution U (t) of a semilinear parabolic PDE (1.1) is assumed, where ps is the stage order of the method This assumption is used to estimate the error of the numerical approximation of the inhomogenous part of the variation of constants formula.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
