$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\xpoint}{\boldsymbol{x}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\It}{\mathcal{I}_t} \newcommand{\setb}[1]{#1^0} % set begin \newcommand{\sete}[1]{#1^{-1}} % set end \newcommand{\setl}[1]{#1^-} \newcommand{\setr}[1]{#1^+} \newcommand{\seti}[1]{#1^i} \newcommand{\Real}{\mathbb{R}} $$

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Index sets in action (2)

A finite difference scheme can with the index set notation be specified as $$ \begin{align*} u^{n+1}_i &= -u^{n-1}_i + 2u^n_i + C^2 \left(u^{n}_{i+1}-2u^{n}_{i}+u^{n}_{i-1}\right), \quad i\in\seti{\Ix},\ n\in\seti{\It}\\ u_i &= 0, \quad i=\setb{\Ix},\ n\in\seti{\It}\\ u_i &= 0, \quad i=\sete{\Ix},\ n\in\seti{\It} \end{align*} $$

Corresponding implementation: 

for n in It[1:-1]:
    for i in Ix[1:-1]:
        u[i] = - u_2[i] + 2*u_1[i] + \ 
               C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1])
    i = Ix[0];  u[i] = 0
    i = Ix[-1]; u[i] = 0

Program wave1D_dn.py

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