Estimates of variable stepsize Runge-Kutta methods for sectorial evolution equations with nonsmooth data
The purpose of the paper is to derive error estimates for variable stepsize Runge-Kutta methods as applied to semilinear evolution equations \[ \dot{u}= Au + f(u),\quad u(0)=u_0\in X^\alpha, \] where \(A\) is a sectorial operator in some Banach space \(X\) which satisfies standard resolvent conditions, \(X^\alpha\), \(\alpha\geq 0\), denote the classical fractional order spaces derived from \(X\), and the nonlinearity \(f: X^\alpha\mapsto X\) is assumed to be globally Lipschitz. The hypotheses made on \(A\) and \(f\) are the following: (A0) \(A\) is a densely defined closed linear operator on \(X\) satisfying the spectral condition \[ \sigma(A)\subset \Sigma_{a,\theta}=\{z\in{\mathbb C}\setminus\{a\}:\;\theta < |\arg(z-a)|\leq \pi\} \] for some real \(a,\theta\) with \(\theta\in (\frac{1}{2}\pi,\pi)\) and the resolvent estimate \[ |(zI-A)^{-1}|\leq \frac{M}{1+|z-a|}\quad \text{whenever} z\not\in\Sigma_{a,\theta} \] for some positive constant \(M\). (F0) For a fixed \(\alpha\in [0,1)\) a mapping \(f:X^\alpha\mapsto X\) is given which satisfies for some \(L\geq 0\) the Lipschitz condition \[ |f(u)-f(\tilde{u})|\leq L|u-\tilde{u}|_\alpha\quad \text{for all} u,\tilde{u}\in X^\alpha. \] The main theorem is the following: Assume that conditions (A0) and (F0) are satisfied and that the Runge-Kutta method is strictly \(A(\theta^c)\)-stable (being \(\theta^c=\pi-\theta\)) and of order \({\mathcal P}\). In addition, assume that the stability function of the Runge-Kutta method, say \(r(z)\), satisfies \(r(\infty)=0\). Let \(T>0\) be arbitrary. Then there exist constants \({\mathcal K}>0\) and \(h_0>0\) such that for all finite stepsize sequences \((h_1,\dots,h_N)\), \(N\geq 1\), with \(H=\sum_{n=1}^N h_n\leq T\) and \(h_\ast=\max_{1\leq n\leq N} h_n\leq h_0\) we have the estimate \[ |U_N - \bar{u}(H)|_\alpha \leq {\mathcal K} \left( \left(\frac{h_\ast}{H}\right)^{\mathcal P} + \frac{h_\ast}{H^\alpha}\log\frac{H}{h_\ast}\right) \] where \(U_N\) stands for the variable stepsize Runge-Kutta solution at time \(H\) and \(\bar{u}(t)\) denotes the solution of the initial value problem \[ \dot{u}= Au + f(u),\quad u(0)=u_0\in X^\alpha. \] The constants \({\mathcal K}\) and \(h_0\) depend on the data \(T, \alpha, |u_0|_\alpha, |f(u_0)|, \theta, a\), on the constants \(M, L\) from (A0), (F0) as well as on the stability function of the Runge-Kutta method.
scientific article; zbMATH DE number 1787709
scientific article