$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Gauss-Legendre rules apply optimized points
- Optimize the location of points to get higher accuracy
- Gauss-Legendre rules (quadrature) adjust points and weights to
integrate polynomials exactly
$$
\begin{align}
M=1&:\quad \bar X_0=-\frac{1}{\sqrt{3}},\
\bar X_1=\frac{1}{\sqrt{3}},\ w_0=w_1=1\\
M=2&:\quad \bar X_0=-\sqrt{\frac{3}{{5}}},\ \bar X_0=0,\
\bar X_2= \sqrt{\frac{3}{{5}}},\ w_0=w_2=\frac{5}{9},\ w_1=\frac{8}{9}
\end{align}
$$
- \( M=1 \): integrates 3rd degree polynomials exactly
- \( M=2 \): integrates 5th degree polynomials exactly
- In general, \( M \)-point rule integrates a polynomial
of degree \( 2M+1 \) exactly.
See numint.py for a large collection of Gauss-Legendre rules.
