{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.


Runge-Kutta contractivity identity

{c}

A Runge--Kutta method is B-stable when it does not increase distances between numerical solutions of a dissipative differential equation. <Algebraic stability of a Runge-Kutta method> is a standard sufficient condition.

= B-stable
{c}
{synonym}


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 Incoming links: B-stability