$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Index set notation
- Tedious to write index sets like \( i=0,\ldots,N_x \) and
\( n=0,\ldots,N_t \)
- Notation not valid if \( i \) or \( n \) starts at 1 instead...
- Both in math and code it is advantageous to use index sets
- \( i\in\Ix \) instead of \( i=0,\ldots,N_x \)
- Definition: \( \Ix =\{0,\ldots,N_x\} \)
- The first index: \( i=\setb{\Ix} \)
- The last index: \( i=\sete{\Ix} \)
- All interior points: \( i\in\seti{\Ix} \), \( \seti{\Ix}=\{1,\ldots,N_x-1\} \)
- \( \setl{\Ix} \) means \( \{0,\ldots,N_x-1\} \)
- \( \setr{\Ix} \) means \( \{1,\ldots,N_x\} \)
