Log In Sign Up

Exponential Runge Kutta time semidiscetizations with low regularity initial data

by   Claudia Wulff, et al.

We apply exponential Runge Kutta time discretizations to semilinear evolution equations d U/ d t=AU+B(U) posed on a Hilbert space Y. Here A is normal and generates a strongly continuous semigroup, and B is assumed to be a smooth nonlinearity from Y_ℓ = D(A^ℓ) to itself, and ℓ∈ I ⊆ [0,L], L ≥ 0, 0,L ∈ I. In particular the semilinear wave equation and nonlinear Schrödinger equation with periodic boundary conditions or posed on R^d fit into this framework. We prove convergence of order O(h^min(ℓ,p)) for non-smooth initial data U^0∈ Y_ℓ, where ℓ >0, for a method of classical order p. We show in an example of an exponential Euler discretization of a linear evolution equation that our estimates are sharp, and corroborate this in numerical experiments for a semilinear wave equation. To prove our result we Galerkin truncate the semiflow and numerical method and balance the Galerkin truncation error with the error of the time discretization of the projected system. We also extend these results to exponential Rosenbrock methods.


page 1

page 2

page 3

page 4


A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation

We consider a numerical approximation of a linear quadratic control prob...

An energy-based summation-by-parts finite difference method for the wave equation in second order form

We develop an energy-based finite difference method for the wave equatio...

A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise

We consider a nonlocal evolution equation representing the continuum lim...

A general framework of low regularity integrators

We introduce a new general framework for the approximation of evolution ...