We study semilinear evolution equations $$ \frac{\mathrm dU}{\mathrm dt}=AU+B(U)$$dUdt=AU+B(U) posed on a Hilbert space $$\mathcal Y$$Y, where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from $$\mathcal Y_\ell = D(A^\ell )$$Yℓ=D(Aℓ) to itself, and $$\ell \in I \subseteq [0,L]$$ℓ∈I⊆[0,L], $$L \ge 0$$L≥0, $$0,L \in I$$0,L∈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 $$O(h^{p\ell /(p+1)})$$O(hpℓ/(p+1)) for non-smooth initial data $$U^0\in \mathcal Y_\ell $$U0∈Yℓ, where $$\ell \le p+1$$ℓ≤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.

中文翻译：

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具有非光滑数据的半线性偏微分方程的 Runge-Kutta 时间半离散化

我们研究了在希尔伯特空间 $$\mathcal Y$$Y 上提出的半线性演化方程 $$ \frac{\mathrm dU}{\mathrm dt}=AU+B(U)$$dUdt=AU+B(U)，其中 A 是正规的并生成强连续半群，B 是从 $$\mathcal Y_\ell = D(A^\ell )$$Yℓ=D(Aℓ) 到自身的平滑非线性，以及 $$\ell \in I \subseteq [0,L]$$ℓ∈I⊆[0,L], $$L \ge 0$$L≥0, $$0,L \in I$$0,L∈I。特别是具有周期性、Neumann 和 Dirichlet 边界条件的一维半线性波动方程和非线性薛定谔方程适合该框架。我们使用 A 稳定 Runge-Kutta 方法在时间上离散演化方程，保留连续空间，并证明 $$O(h^{p\ell /(p+1)})$$O(hpℓ/ (p+1)) 对于非平滑初始数据 $$U^0\in \mathcal Y_\ell $$U0∈Yℓ，其中 $$\ell \le p+1$$ℓ≤p+1，对于 a经典阶 p 的方法，扩展 Brenner 和 Thomée 对线性系统的结果。我们的方法是将半流和数值方法投影到谱伽辽金近似，并平衡投影误差与投影系统的时间离散化误差。数值实验表明我们的估计是尖锐的。