Absolute stability regions

Apply a one-step method to $y'=\lambda y$ and write the result as $Y_{n+1}=R(z)Y_n$, where $z=h\lambda$. The exact amplification is $e^z$. We use the closed stability region defined by $|R(z)|\le1$ for bounded scalar iterates; its strict interior $|R(z)|<1$ gives decay to zero from a nonzero starting value. A boundary point with modulus one need not damp at all, even when the exact differential equation decays.

Forward Euler replaces the exponential by one plus z, so its region is the disk centered at minus one with radius one. Backward Euler has amplification one divided by one minus z, with a pole at z=1; its boundedness region is outside or on the disk centered at one with radius one. The location of the left half-plane in these pictures explains their different behavior on rapid decay. Neither diagram directly measures how closely the numerical factor approximates the exact exponential.

For a negative real rate, forward Euler decays only when the positive step is strictly below two divided by the rate magnitude. At the endpoint the sign alternates with unchanged magnitude. For a complex rate with negative real part, expanding the squared modulus yields a step restriction involving both real decay and oscillation frequency. Rapid oscillation can make this restriction severe even when the damping rate is modest. A stable step may still give a badly distorted oscillation angle.

For a diagonalizable constant matrix, transforming into eigenvectors reduces the recurrence to scalar amplification of each eigenvalue. The conditioning of that eigenvector transformation affects the norm bound. A nonnormal matrix may exhibit substantial transient growth even when every scalar amplification lies strictly inside the disk. On the boundary, Jordan blocks can generate polynomial growth despite eigenvalues of modulus one. Thus a spectral plot is an important component of matrix stability analysis, not a universal normwise certificate.

Suppose a stable factor has magnitude one half while the exact factor has magnitude one tenth. Both decay, but they describe different physical relaxation. Likewise a factor on the unit circle can preserve amplitude while advancing phase incorrectly. The desired resolution of slow behavior, phase, or a quantity of interest must therefore be checked separately. Absolute stability tells us which steps can avoid a particular numerical instability; consistency and an error assessment tell us whether the resulting trajectory answers the original question.

For a nonlinear equation, the Jacobian along a trajectory supplies a local linear diagnostic. Its variation, the geometry of perturbations, and the changing step all matter. One cannot freeze one initial Jacobian and infer that its eigenvalues certify the full nonlinear simulation. A practical investigation combines scalar regions with refinement checks, invariant or residual monitoring, and knowledge of the model's time scales. Those checks explain a failure more effectively than treating every oscillating numerical curve as physical behavior.