26 - Dahlquist’s Equivalence Theorem

Dahlquist's Equivalence Theorem

Consider consistent linear $k$-step method with consistent starting values

Root Condition (zero-stability) is necessary and sufficient for convergence so

$$\underset{h\to 0}{\lim }{\mathbf{y}}_N=\mathbf{y}(X)\text{ where }N=(X-x_0)/h$$

Moreover if

$$\tau (h)=O(h^p)\text{ and }||\mathbf{y}(x_s)-{\eta }_s(h)||=O(h^p)\text{ for }s=0,\cdots ,k-1$$

Then

$$\underset{0\le n\le N}{\max }||\mathbf{y}(x_n)-{\mathbf{y}}_n||=O(h^p)$$