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Applying a linear multistep method to the Dahlquist test equation with $z=h\lambda$ produces a constant-coefficient recurrence whose modes satisfy $P_z(w)=0$. Absolute stability requires the resulting roots of a polynomial to lie in the unit disk, with unit-modulus roots simple. Thus multistep stability depends on every recurrence mode, rather than a single one-step stability function.

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 $a$.

The classical four-stage explicit Runge-Kutta method usesIt satisfies the fourth-order conditions for a Runge-Kutta method. Its polynomial stability function excludes A-stability, while $|R(iq){|}^2=1-q^6/72+q^8/576$ gives the imaginary-axis stability interval $|q|\le 2\sqrt{2}$.

The Dahlquist test equation is the scalar linear initial-value problem $y^{\prime }=\lambda y$. A one-step method reduces it to $y_{n+1}=R(h\lambda )y_n$, thereby defining its stability function $R$.

The Euclidean logarithmic norm is the largest eigenvalue of a matrix's Hermitian part. Differentiating $\Vert y{\Vert }_2^2$ along $y^{\prime }=Ay$ gives $\Vert e^{tA}{\Vert }_2\le e^{t{\mu }_2(A)}$. It is the least exponent for such a bound with prefactor one, as a first-order expansion at zero shows. It equals the spectral abscissa for a normal matrix, but need not do so otherwise. A scalar rational stability function does not generally inherit a bound by its value at this real number.

The two-stage Gauss--Legendre Runge-Kutta method uses nodes $c_{1,2}=1/2\mp \sqrt{3}/6$, weights $b_1=b_2=1/2$ and coefficients $a_{11}=a_{22}=1/4$, $a_{12}=1/4-\sqrt{3}/6$, $a_{21}=1/4+\sqrt{3}/6$. The Butcher order conditions give order four. Its stability function is the displayed rational function, which is A-stable. The algebraic stability matrix is zero, so positive weights also give nonlinear contractivity for dissipative vector fields.

A Lobatto IIIA method collocates at the Lobatto quadrature nodes, including both endpoints of the time interval. Its three-stage version hasThe Butcher order conditions give order four. Its stability function iswhich is A-stable but not L-stable. It is not algebraically stable: the first diagonal entry of the defining matrix is $2b_1{a}_{11}-b_1^2=-1/36$. This distinguishes linear A-stability from nonlinear algebraic 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.