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With $h=\delta t>0$, the central finite difference converts the decay equation toThe continuous ordinary differential equation has the general solution $x(t)=C{e}^{-Kt}$. The discrete linear recurrence is second order and requires two initial values, such as $x_0,x_1$. Trying $x_n={\rho }^n$ gives the characteristic equation of a linear recurrence ${\rho }^2+2Kh\rho -1=0$. Its roots aresoThere are no repeated roots, and the same formula extends to negative integer $n$ if required.

To demonstrate the continuum limit, put $a_h=\mathrm{arsinh}(Kh)$. Then ${\rho }_{+}=e^{-a_h}$ and ${\rho }_{-}=-e^{a_h}$, with $a_h/h\to K$. For $t_n=nh\in [0,T]$,uniformly, since $0\le t_n\le T$. More precisely $a_h/h=K-K^3{h}^2/6+O(h^4)$, giving an $O(h^2)$ error on a fixed interval. Thus $B=0$, $A=C$ selects the physical discrete mode and corresponds to $x_1={\rho }_{+}{x}_0$.

The other root is a parasitic amplification root:For fixed $B\ne 0$, even and odd mesh points approach values differing by $2B{e}^{Kt}$. Equivalently successive mesh values have a nonvanishing alternating jump despite their time separation tending to zero. This mode cannot converge uniformly to a continuous ODE solution; its limiting envelope also grows rather than decays.

For step-dependent initial data the precise result is the centered two-step discretization of exponential decay criterion:Sufficiency follows from the two uniformly bounded exponential envelopes on a fixed interval. Necessity follows from convergence at the adjacent points $0,h$: $x_0\to C$, $x_1-x_0\to 0$, and the coefficient formulas force $B_h\to 0$, $A_h\to C$. In particular a nonzero but vanishing starting error is not excluded. Starting with the exact value $x_1=C{e}^{-Kh}$ gives $B_h=C{K}^3{h}^3/12+O(h^4)$ and still converges on each fixed interval. The assertion about “remaining” solutions therefore concerns a fixed nonzero parasitic amplitude, not every nonzero step-dependent amplitude. The method is nevertheless not damping-stable for arbitrarily long times because $|{\rho }_{-}|>1$.