MATH761

Definition. For grid data $𝐐_{i} = {Q_{i + j} ≅ q (x_{i} + j h), j \in J}$ , a finite difference formula $ℱ_{h}^{k} (𝐐_{i})$ is a $p^{th}$ -order accurate approximation of the $k^{th}$ derivative $q^{(k)} (x_{i})$ if $|| ℱ_{h}^{k} (𝐐_{i}) - q^{(k)} (x_{i}) || = 𝒪 (h^{p})$ .

In[9]:= Dp[f[x],1]

$\frac{f (h + x) - f (x)}{h}$

In[11]:= Series[ Dp[f[x],1] ,{h,0,2}] - f'[x]

$\frac{1}{2} h f^{''} (x) + \frac{1}{6} h^{2} f^{(3)} (x) + O (h^{3})$

In[24]:= Table[ {(Dp[f[i],p] /. f->Q /. h->1)/h,Limit[(Normal[Series[Dp[f[x],p],{h,0,p}]] - f'[x])/h^p, h->0] /. f->q},{p,1,5}] // TableForm

$\begin{array}{cc} \frac{Q_{i + 1} - Q_{i}}{h} & \frac{q^{''} (x)}{2} \\ \frac{- 3 Q_{i} + 4 Q_{i + 1} - Q_{i + 2}}{2 h} & - \frac{1}{3} q^{(3)} (x) \\ \frac{- 11 Q_{i} + 18 Q_{i + 1} - 9 Q_{i + 2} + 2 Q_{i + 3}}{6 h} & \frac{1}{4} q^{(4)} (x) \\ \frac{- 25 Q_{i} + 48 Q_{i + 1} - 36 Q_{i + 2} + 16 Q_{i + 3} - 3 Q_{i + 4}}{12 h} & - \frac{1}{5} q^{(5)} (x) \\ \frac{- 137 Q_{i} + 300 Q_{i + 1} - 300 Q_{i + 2} + 200 Q_{i + 3} - 75 Q_{i + 4} + 12 Q_{i + 5}}{60 h} & \frac{1}{6} q^{(6)} (x) \end{array}$

In[25]:=