structural:reformatting
B-stability

The A-stability concept for the solution of differential equations is related to the linear autonomous equation y ′ = λ y {\displaystyle y'=\lambda y}. Dahlquist proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition. The corresponding concepts were defined as G-stability for multistep methods (and the related one-leg methods) and B-stability (Butcher, 1975) for Runge–Kutta methods. A Runge–Kutta method applied to the non-linear system y ′ = f (y) {\displaystyle y'=f(y)}, which verifies ⟨ f (y) − f (z), y − z ⟩ ≤ 0 {\displaystyle \langle f(y)-f(z),\ y-z\rangle \leq 0}, is called B-stable, if this condition implies ‖ y n + 1 − z n + 1 ‖ ≤ ‖ y n − z n ‖ {\displaystyle \|y_{n+1}-z_{n+1}\|\leq \|y_{n}-z_{n}\|} for two numerical solutions.

Let B {\displaystyle B}, M {\displaystyle M} and Q {\displaystyle Q} be three s × s {\displaystyle s\times s} matrices defined by B = diag (b 1, b 2, …, b s), M = B A + A T B − b b T, Q = B A − 1 + A − T B − A − T b b T A − 1. {\displaystyle B=\operatorname {diag} (b_{1},b_{2},\ldots,b_{s}),\,M=BA+A^{T}B-bb^{T},\,Q=BA^{-1}+A^{-T}B-A^{-T}bb^{T}A^{-1}.} A Runge–Kutta method is said to be algebraically stable if the matrices B {\displaystyle B} and M {\displaystyle M} are both non-negative definite. A sufficient condition for B-stability is: B {\displaystyle B} and Q {\displaystyle Q} are non-negative definite.