Runge–Kutta time semidiscretizations of semilinear PDEs with non-smooth data

详细信息    查看全文

• 作者：Claudia Wulff ; Chris Evans
• 刊名：Numerische Mathematik
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：134
• 期：2
• 页码：413-440
• 全文大小：721 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245
• 卷排序：134

文摘

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 \(O(h^{p\ell /(p+1)})\) for non-smooth initial data \(U^0\in \mathcal Y_\ell \), 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.Mathematics Subject Classification65J0865J1565M1265M15