 Even-reflection closure for Neumann diffusion

For the heat equation with homogeneous Neumann boundary conditions at both ends of a uniform grid, even reflection supplies ghost values for a centered finite difference method. A fourth-order centered second derivative needs two ghost values at each end, obtained with the displayed rule for $j=1,2$. For a sufficiently smooth compatible solution, differentiating the Neumann boundary conditions in time and using the heat equation makes every odd spatial derivative vanish at the endpoints. Its even extension therefore retains the smoothness required by the fourth-order stencil; arbitrary incompatible initial data need care at the initial boundary corners.

The discrete cosine modes diagonalize this reflected spatial operator, with eigenvalues $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]$ and $\theta =j\pi /N$. Its most negative eigenvalue is $-16/(3h^2)$, so the forward Euler method is stable in the corresponding trapezoid-weighted norm when $\Delta t/h^2\le 3/8$. The Crank-Nicolson diffusion scheme is stable for every time step for this closure, but its amplification factor approaches $-1$ on highly damped spatial modes, allowing sign-alternating numerical transients. The ghost values are defined here as auxiliary samples outside the physical interval, rather than additional unknown initial data.