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Neumann condition and a variable coefficient
Consider \partial u/\partial x=0 at x=L=N_x\Delta x :
\frac{u_{i+1}^{n} - u_{i-1}^n}{2\Delta x} = 0\quad u_{i+1}^n = u_{i-1}^n,
\quad i=N_x
Insert u_{i+1}^n=u_{i-1}^n in the stencil
(30)
for i=N_x and obtain
u^{n+1}_i \approx
- u_i^{n-1} + 2u_i^n + \left(\frac{\Delta x}{\Delta t}\right)^2
2q_{i}(u_{i-1}^n - u_{i}^n) + \Delta t^2 f^n_i
(We have used q_{i+\half} + q_{i-\half}\approx 2q_i .)
Alternative: assume dq/dx=0 (simpler).
