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Expanding the symmetric spatial stencil by the Taylor theorem givesConsistency of a numerical method and cancellation of the $h^2$ term require $\alpha +2\beta +2\gamma =0$, $\beta +4\gamma =1$, and $\beta +16\gamma =0$. Solving this linear system givesThe resulting fourth-order centered second derivative hasThe nonzero next coefficient proves that the highest spatial order of a numerical method is four for this three-parameter stencil.

For the method of lines in part (a), the Fourier symbol of the spatial operator isEach Fourier mode therefore evolves by $\hat{U}(\theta ,t)=e^{t{\lambda }_h(\theta )}\hat{U}(\theta ,0)$, whose multiplier has modulus at most one. The discrete Parseval identity givesThus the fourth-order centered second derivative gives an unconditionally stable semidiscrete evolution in the L2 norm. No time integration method has yet been chosen, so no time-step restriction belongs to this conclusion.

An arbitrary L2 function does not have well-defined point samples. For the stated initial data one may use cell-average projection, $U_m(0)=h^{-1}{\int }_{x_m-h/2}^{x_m+h/2}{u}_0(x)dx$. The Cauchy-Schwarz inequality gives $h{\sum }_m|U_m(0){|}^2\le \Vert u_0{\Vert }_{L^2}^2$, providing a bounded initialization for the stability estimate.