MATH566 Lesson 18: Differentiation matrices

Overview

• Differentiation at multiple grid points

• Differentiation matrix as the discrete version of the differentiation operator

Derivatives at multiple grid points

• Consider $f:ℝ\to ℝ$, sampled at $\{x_0,x_1,\dots ,x_n\}$, $f_j\equiv f\left(x_j\right)$, $𝒇={\left[\begin{array}{lll}{f}_0& \dots & f_n\end{array}\right]}^T$

• Evaluate $f^{\text{'}}$ at points $\{x_0^{\text{'}},x_1^{\text{'}},\dots ,x_m^{\text{'}}\}$, $f_j^{\text{'}}\cong f^{\text{'}}\left(x_j^{\text{'}}\right)$, ${𝒇}^{\text{'}}={\left[\begin{array}{lll}{f}_0^{\text{'}}& \dots & f_m^{\text{'}}\end{array}\right]}^T$

• Finite difference formulas: derivative approximations as linear combinations

  ${\displaystyle {𝒇}^{\text{'}}=𝑫𝒇,𝑫=\left[d_{ij}\right],d_{ij}=\frac{l_j^{\text{'}}\left(x_i^{\text{'}}\right)}{l_j\left(x_j\right)},l_j\left(t\right)=\prod _{k=0}^n{}^{\text{'}}(t-x_k)}$ ${\displaystyle p\left(t\right)=\sum _{j=0}^n{f}_j{\ell }_j\left(t\right)\Rightarrow p^{\text{'}}\left(t\right)=\sum _{j=0}^n{f}_j{\ell }_j^{\text{'}}\left(t\right)}$ ${\displaystyle f^{\text{'}}\left(x_i^{\text{'}}\right)\cong \sum _{j=0}^n{f}_j{\ell }_j^{\text{'}}\left(x_i^{\text{'}}\right)=\sum _{j=0}^n{f}_j\frac{l_j^{\text{'}}\left(x_i^{\text{'}}\right)}{l_j\left(x_j\right)}}$

Centered first order differentiation matrix on uniform grid

• $f:ℝ\to ℝ$, sampled at $\{x_0,x_1,\dots ,x_n\}$, $f_j\equiv f\left(x_j\right)$, $𝒇={\left[\begin{array}{lll}{f}_0& \dots & f_n\end{array}\right]}^T$

• Evaluate $f^{\text{'}}$ at points $\{x_1,\dots ,x_{n-1}\}$, $f_j^{\text{'}}\cong f^{\text{'}}\left(x_j\right)$, ${𝒇}^{\text{'}}={\left[\begin{array}{lll}{f}_1^{\text{'}}& \dots & f_{n-1}^{\text{'}}\end{array}\right]}^T$

• Centered finite difference $f_j^{\text{'}}\cong (f_{j+1}-f_{j-1})/\left(2h\right)$

  ${\displaystyle {𝒇}^{\text{'}}={𝑫}_{2h}𝒇\Rightarrow \left[\begin{array}{l}{f}_1^{\text{'}}\\ f_2^{\text{'}}\\ ⋮\\ f_{n-1}^{\text{'}}\end{array}\right]=\frac{1}{2h}\left[\begin{array}{llllll}-1& 0& 1& & & \\ & -1& 0& 1& & \\ & & \ddots & \ddots & \ddots & \\ & & & -1& 0& 1\end{array}\right]\left[\begin{array}{l}{f}_0\\ f_1\\ f_2\\ ⋮\\ f_n\end{array}\right],}$

• Sampling at half interval: $\{x_0,x_{1/2},\dots ,x_n\}$, $f_j\equiv f\left(x_j\right)$, $𝒇={\left[\begin{array}{lll}{f}_0& \dots & f_n\end{array}\right]}^T$

  ${\displaystyle {𝒇}^{\text{'}}={𝑫}_h𝒇\Rightarrow \left[\begin{array}{l}{f}_1^{\text{'}}\\ f_2^{\text{'}}\\ ⋮\\ f_{n-1}^{\text{'}}\end{array}\right]=\frac{1}{h}\left[\begin{array}{llllll}-1& 0& 1& & & \\ & -1& 0& 1& & \\ & & \ddots & \ddots & \ddots & \\ & & & -1& 0& 1\end{array}\right]\left[\begin{array}{l}{f}_{1/2}\\ f_1\\ f_{3/2}\\ ⋮\\ f_{n-1/2}\end{array}\right],}$

Differentiation matrix = discrete differentiation operator

• Calculus: differentiation operator $D=d/dt$, higher order derivatives:

  ${\displaystyle f^{\left(k\right)}\left(t\right)=\frac{d^k}{d{t}^k}f\left(t\right)=D^{k}f=D\left(D\left(\dots D\left(f\right)\right)\right)}$

• Similar properties hold for differentation matrices, e.g., ${𝒇}^{\text{'}\text{'}}={𝑫}_h{𝑫}_h𝒇$

  ${\displaystyle {𝑫}_h{𝑫}_h=\frac{1}{h^2}\left[\begin{array}{llllll}-1& 0& 1& & & \\ & -1& 0& 1& & \\ & & \ddots & \ddots & \ddots & \\ & & & -1& 0& 1\end{array}\right]\left[\begin{array}{llllll}-1& 0& 1& & & \\ & -1& 0& 1& & \\ & & \ddots & \ddots & \ddots & \\ & & & -1& 0& 1\end{array}\right]\Rightarrow }$ ${\displaystyle {𝑫}_h^2=\frac{1}{h^2}\left[\begin{array}{lllllll}1& 0& -2& 0& 1& & \\ & 1& 0& -2& 0& 1& \\ & & \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right]\Rightarrow }$ ${\displaystyle f_n^{\text{'}\text{'}}\cong \frac{f_{n-1}-2f_n+f_{n+1}}{h^2}}$

Order of accuracy verification

• Applying ${𝑫}_h^2$ to $𝒇$ leads to

  ${\displaystyle f_n^{\text{'}\text{'}}\cong \frac{f_{n-1}-2f_n+f_{n+1}}{h^2}}$

• Taylor series analysis

  ${\displaystyle \begin{array}{l}{f}_{n+1}=f\left(x_{n+1}\right)=f_n+f_n^{\text{'}}\cdot h+\frac{1}{2}{f}_n^{{}^{\text{'}\text{'}}}\cdot h^2+\frac{1}{6}{f}_n^{\text{'}\text{'}\text{'}}{h}^3+\cdots \\ f_{n-1}=f\left(x_{n-1}\right)=f_n-f_n^{\text{'}}\cdot h+\frac{1}{2}{f}_n^{{}^{\text{'}\text{'}}}\cdot h^2-\frac{1}{6}{f}_n^{\text{'}\text{'}\text{'}}{h}^3+\cdots \end{array}}$ ${\displaystyle \frac{1}{h^2}(f_{n-1}-2f_n+f_{n+1})=f_n^{\text{'}\text{'}}+\frac{1}{12}{f}_n^{\left(\mathrm{iv}\right)}{h}^2=f_n^{\text{'}\text{'}}+𝒪\left(h^2\right)}$