From J.C. Strikwerda's book Finite Difference Schemes and Partial Differential Equations (SIAM), p. 57:
2.2.5. Show that the scheme $$
\frac{v_m^{n+1} – v_m^n}{k} + a\frac{v_{m+2}^{n}- 3 v_{m+1}^{n} + 3 v_m^n – v_{m-1}^{n}}{h^3} = f_m^n
$$
is consistent with the equation
$$ \left[ u_t+au_{xxx}=f \right] \tag{2.2.15}$$
and, if $\nu = kh^{-3}$ is constant, then it is stable when $0\leq a\nu \leq 1/4$.
Actually, I know how to solve this problem when $f=0$, consistency condition ($Pu-p_{k,h} u$ converges to zero as $k,h$ tend zero).
- How can I deal with $f(t_n, x_m)$ in this case?! I think we need grid function instead of $f(t_n, x_m)$. can I suppose $f(t_n, x_m) =g^n \exp( imhs)$?
- How to solve the stability part by using von Neumann analysis? Could you help me? Thank you in advance.
Best Answer
For the consistency part, expand $v_m^{n+1}$ and $v_{m+M}^{n}$ as Taylor series about $v_{m}^{n} \simeq u(mh,nk)$ to be inserted in the scheme. As discussed in Definition 1.4.2 of the book by Strikwerda, the source $f$ does not matter in the definition of consistency. Thus, $$ \frac{v_{m}^{n+1} - v_{m}^{n}}{k} \simeq (u_t)_m^n + O(h,k) $$ $$ \frac{v_{m+2}^{n} - 3 v_{m+1}^{n} + 3 v_{m}^{n} - v_{m-1}^{n}}{h^3} \simeq (u_{xxx})_m^n + O(h,k) $$ proves that the scheme is consistent.
As usual, the Von Neumann analysis consists in a Fourier transformation of the scheme (see Sec. 2.2 of the book by Strikwerda), and the source $f$ can be ignored. Under the periodicity assumption, we have $$ v_m^{n} = \frac{1}{\sqrt{2\pi}}\int_{-\pi/h}^{\pi/h} \text{e}^{\text{i}mh\xi}\, \hat{v}^n(\xi) \, d\xi $$ to be injected in the scheme. This step defines the amplification factor $g(\theta)$ to be determined, which is defined in such a way that $\hat{v}^{n+1}(\xi) = g(h\xi) \, \hat{v}^{n}(\xi).$ Neumann stability imposes $|g(\theta)| \leq 1$.