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Inserting $$ u^n_q = A^n e^{ikq\Delta x}$$ leads to $$ A = 1 -4F\sin^2\left( \frac{k\Delta x}{2}\right),\quad F = \frac{\dfc\Delta t}{\Delta x^2}\mbox{ (mesh Fourier number)} $$
The complete numerical solution is $$ u^n_q = (1 -4F\sin^2 p)^ne^{ikq\Delta x},\quad p = k\Delta x/2 $$
Key spatial discretization quantity: the dimensionless \( p=\half k\Delta x \)
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