Algebraic stability of a Runge-Kutta method

A Runge-Kutta method is algebraically stable when

b_{i} \geq 0

and the matrix

M

with

M_{ij} = b_{i} a_{ij} + b_{j} a_{ji} - b_{i} b_{j}

is positive semidefinite. Algebraic stability implies contractivity for dissipative differential equations.

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- Runge-Kutta contractivity identity Algebraic stability of a Runge-Kutta method

Runge-Kutta contractivity identity

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} + 2 h \sum_{i} b_{i} Re ⟨ D_{i}, F_{i} ⟩ - h^{2} \sum_{i, j} m_{ij} Re ⟨ F_{i}, F_{j} ⟩ .

This identity proves that algebraic stability implies B-stability for a dissipative vector field.

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