h_{x} = 1/ (N + 1)

and

J = 2 N + 1

. The Dirichlet discrete Laplacian has the eigenvectors

v_{j} (m) = sin \frac{jπ ( m + N + 1 )}{2 N + 2}, j = 1, \dots, 2 N + 1,

(1)

and its negative has the eigenvalues

λ_{j}^{h} = \frac{4}{h _{x}^{2}} sin^{2} \frac{jπ}{4 ( N + 1 )} .

(2)

The discrete sine transform therefore reduces the method of lines equations to

q_{j}^{''} = (α - λ_{j}^{h}) q_{j}

. Each coefficient is an oscillator, a linear function, or a hyperbolic function according to the sign of

λ_{j}^{h} - α

. A finite collection of oscillators is bounded in every fixed-grid norm. A zero-frequency mode can grow linearly, and a negative-frequency-square mode can grow exponentially. Consequently,

α < 4 (N + 1)^{2} sin^{2} \frac{π}{4 ( N + 1 )}

(3)

is necessary and sufficient for all-time boundedness of a semidiscrete reaction wave equation on this grid. Equality allows bounded displacement only when the initial velocity has zero projection onto the first eigenvector; above it, the growing components must be absent in every unstable mode. A strictly positive spectral gap also gives a time-independent displacement bound uniform over grids whose gaps are bounded below.

The discrete threshold is strictly smaller than

π^{2} /4

, because

sin s < s

for

s > 0

, and tends to

π^{2} /4

N \to \infty

. A coarse grid can therefore introduce long-time growth into a continuum problem with

α < π^{2} /4

. This is distinct from displacement stability of a symmetric semidiscrete wave equation on fixed finite time intervals. As in part (a), if existence of the printed limit is required for every initial datum, no $α$ qualifies: any one mode admits oscillatory, secular, or growing initial data, depending on its coefficient.