Gauss-Legendre method

The $s$-stage Gauss--Legendre Runge--Kutta method collocates at the Gauss--Legendre quadrature nodes on $[0,1]$. It is symmetric, A-stable, and has order $2s$.

Choose $c_1,\dots ,c_s$ as the polynomial roots of $P_s(2c-1)$ in $(0,1)$, and form the Lagrange interpolation polynomials ${\ell }_j(t)={\prod }_{m\ne j}(t-c_m)/(c_j-c_m)$. The Gauss collocation method has $a_{ij}={\int }_0^{c_i}{\ell }_j(t)dt$ and $b_j={\int }_0^1{\ell }_j(t)dt$. This gives explicit coefficient formulas after finding polynomial roots: if ${\ell }_j(t)={\sum }_r{d}_{jr}{t}^r$, these integrals are ${\sum }_r{d}_{jr}{c}_i^{r+1}/(r+1)$ and ${\sum }_r{d}_{jr}/(r+1)$. Gaussian quadrature gives order $2s$.

The two-stage Gauss--Legendre Runge-Kutta method uses nodes $c_{1,2}=1/2\mp \sqrt{3}/6$, weights $b_1=b_2=1/2$ and coefficients $a_{11}=a_{22}=1/4$, $a_{12}=1/4-\sqrt{3}/6$, $a_{21}=1/4+\sqrt{3}/6$. The Butcher order conditions give order four. Its stability function is the displayed rational function, which is A-stable. The algebraic stability matrix is zero, so positive weights also give nonlinear contractivity for dissipative vector fields.