= Runge-Kutta contractivity identity
{c}
For differences $D_i$ between corresponding stages and differences $F_i$ between their vector fields, a Runge--Kutta step satisfies
$$
\|d_{n+1}\|^2=\|d_n\|^2+2h\sum_i b_i\operatorname{Re}\langle D_i,F_i\rangle-h^2\sum_{i,j}m_{ij}\operatorname{Re}\langle F_i,F_j\rangle.
$$
To derive the identity, expand the squared <norm> of $d_{n+1}=d_n+h\sum_i b_iF_i$ and substitute $d_n=D_i-h\sum_j a_{ij}F_j$ in the linear term. Symmetrizing the double sum gives the displayed coefficients $m_{ij}$. For a <dissipative vector field>, the first sum is nonpositive when $b_i\geq0$, and the second sum is nonnegative when the <matrix> $(m_{ij})$ is a <positive semidefinite matrix>. Thus <algebraic stability> implies <B-stability>.
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