Given an ODE $\dot x_t = f(x_t)$ for $t \in [0,T]$ and a non-linear approximation scheme
$$\bar x_{t_{k+1}}^{\Delta} = P(\bar x_{t_k}^{\Delta}, f, \Delta)$$
where $t_k = k \Delta, k \in \mathbb{N}$ and $\Delta$ is the step size.
I'd like to test numerically the order of convergence of $P$.
Since I assume that the exact solution is not known I can't directly check how the error $e_{\Delta} = |\bar x_T^{\Delta} – x_T|$ shrinks for decreasing values of $\Delta$, but I was wondering if there exists any method to bound it using the increment $$|\bar x_T^{\Delta} – \bar x_T^{\delta}|$$
Best Answer
If the method has order $p$, then $x^Δ_t=x(t)+c(t)Δ^p+O(Δ^{p+1})$. There are 3 unknowns in this formula, $x(t)$, $c(t)$ and $p$. Thus take 3 step sizes and solve the system $$ x^{2Δ}_t-x^Δ_t=(2^{p}-1)c(t)Δ^p+O(Δ^{p+1})\implies \frac{x^{4Δ}_t-x^{2Δ}_t}{x^{2Δ}_t-x^Δ_t}=2^p+O(Δ) $$