Before revision:  2}\leq \alpha{1-\alpha}\|{err}^{k}\|_{{\mathbf W}, 2} provided that the time-integrator } (z), we prove that the decay rate of the global error is bounded by \alpha/(1-\alpha), provided the method } is stable, where k is the iteration index and for any vector v the norm \|v\|_{{\mathbf W, 2} is defined by \|v\|_{{\mathbf W}, 2}=\|{\mathbf W}v\|_2 with } \mathbf W being a matrix depending on the space discretization matrix only.
After revision:  2}\leq \alpha{1-\alpha}\|{err}^{k}\|_{{\mathbf W}, 2} provided that the time-integrator } method (such as the Runge-Kutta methods) with stability function \mathcal{R(z), we prove that the decay rate of the global error is bounded by \alpha/(1-\alpha), provided the method } is stable, where k is the iteration index and for any vector v the norm \|v\|_{{\mathbf W, 2} is defined by \|v\|_{{\mathbf W}, 2}=\|{\mathbf W}v\|_2 with } \mathbf W being a matrix depending on the space discretization matrix only.