Parasitic amplification root

An amplification root of a multistep method is parasitic when it represents an additional numerical mode rather than the principal mode approximating the exact differential-equation flow. For a consistent method the principal root starts at one as the step tends to zero, while other roots can start elsewhere. A simple parasitic root on the unit circle is compatible with zero-stability, but its motion outside the circle for a negative test parameter can destroy absolute stability. Fixed-time convergence and all-time damping must be distinguished.

Replacing the derivative in $x^{\prime }=-Kx$ by a central finite difference gives $x_{n+1}+2Kh{x}_n-x_{n-1}=0$. Its principal root is ${\rho }_{+}=e^{-\mathrm{arsinh}(Kh)}$, while the parasitic amplification root is ${\rho }_{-}=-e^{\mathrm{arsinh}(Kh)}$. On a fixed time interval, $A_h{\rho }_{+}^n+B_h{\rho }_{-}^n$ converges uniformly to $C{e}^{-Kt}$ exactly when $A_h\to C$ and $B_h\to 0$. A fixed nonzero parasitic amplitude alternates between two incompatible limits, but a vanishing starting error can converge despite the root's modulus exceeding one. This distinguishes fixed-time numerical convergence from long-time absolute stability.