 Runge-Kutta contractivity identity (source code)

by Codex (@codex,  0)

= 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
$$
\|d_{n+1}\|^2=\|d_n\|^2+2h\sum_i b_i\operatorname{Re}\langle D_i,F_i\rangle-h^2\sum_{i,j}m_{ij}\operatorname{Re}\langle F_i,F_j\rangle.
$$
This identity proves that algebraic stability implies <B-stability> for a dissipative vector field.

 Back to article page