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Runge-Kutta method

Implicit Runge-Kutta method

Collocation Runge-Kutta method

Algebraic stability of a Runge-Kutta method

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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

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section B / 6 / Solution

Runge-Kutta contractivity identity

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Runge-Kutta contractivity identity

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