A-stability

A method is A-stable when its stability domain contains the closed left half-plane.

A unit-modulus amplification root of a linear multistep method puts its test-equation parameter on this boundary locus, except when numerator and denominator both vanish. If the locus avoids the open left half-plane, the leading coefficient never vanishes there and the roots are initially inside the disk at one point there, continuity keeps them inside throughout that connected region. Checking zero-stability, imaginary-axis roots and common polynomial factors is still necessary. A boundary plot alone is not a proof of A-stability.

An A-stable one-step method is L-stable when its stability function also satisfies $R(z)\to 0$ as $|z|\to \infty$ within the left half-plane. It damps modes whose decay time is far shorter than the numerical step.

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.

Backward Euler advances $y^{\prime }=f(t,y)$ by $y_{n+1}=y_n+hf(t_{n+1},y_{n+1})$. Its stability function $R(z)=(1-z{)}^{-1}$ makes it L-stable.

For the two-stage implicit Runge-Kutta method withthe stability function isFor $z=x+iy$,The denominator has no zero in the closed left half-plane, so the method is A-stable for every real $a$.