Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 68 / 1 / b

Use the Gauss collocation coefficient construction. Find the $\nu$ polynomial roots $c_1,\dots ,c_{\nu }$ of the shifted Legendre polynomial $P_{\nu }(2c-1)$ in $(0,1)$, and formThen setThese are explicit polynomial integrals. For example, if ${\ell }_j(s)={\sum }_{r=0}^{\nu -1}{d}_{jr}{s}^r$, calculate $a_{ij}={\sum }_r{d}_{jr}{c}_i^{r+1}/(r+1)$ and $b_j={\sum }_r{d}_{jr}/(r+1)$. The resulting implicit Gauss collocation method has order $2\nu$; its Gaussian quadrature nodes integrate polynomials through degree $2\nu -1$ exactly. The one-stage example is the implicit midpoint rule, with $c_1=a_{11}=1/2$ and $b_1=1$. No search over nonlinear Runge-Kutta order conditions systems is needed.