23 - Runge-Kutta Solution of the Dahlquist Test Equation

Runge–Kutta Solution of the Dahlquist Test Equation

Let $\{y_k{\}}_{k\in \mathbb{N}}$ be Runge-Kutta solution to Dahlquist test equation obtained with time step $h>0$
Then

$$y_k=S(zh{)}^k$$

Proof

By direct computation then

$$y_1=\mathbf{\Psi }(0,1,h,z)=\mathbf{\Psi }(0,1,1,zh)=S(zh)$$

And

$$y_2=\mathbf{\Psi }(0,y_1,h,z)=\mathbf{\Psi }(0,1,1,zh)y_1=S(zh)y_1=S(zh{)}^2$$

Hence

$$y_k=S(h\lambda {)}^k$$