= Centered two-step discretization of exponential decay
{title2=$\rho_\pm=-Kh\pm\sqrt{1+K^2h^2}$}
Replacing the derivative in $x'=-Kx$ by a <central finite difference> gives $x_{n+1}+2Khx_n-x_{n-1}=0$. Its principal root is $\rho_+=e^{-\operatorname{arsinh}(Kh)}$, while the <parasitic amplification root> is $\rho_-=-e^{\operatorname{arsinh}(Kh)}$. On a fixed time interval, $A_h\rho_+^n+B_h\rho_-^n$ converges uniformly to $Ce^{-Kt}$ exactly when $A_h\to C$ and $B_h\to0$. 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>.
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