Finite Difference Schemes

Take $U_t=bU_{xx}+f(x,t)$. Let $u^{n}_m=U(nk, mh)=U(t_k, x_m)$

Let $\delta_x$ be the difference operator, such that

$$\delta_xu^n_m=u^n_{m+\frac12}-u^n_{m-\frac12}$$

similarly, $\delta_tu^n_m=u^{n+\frac12}_m-u^{n-\frac12}_m.$

Exercise: Show that $\delta_x^2 u^n_m=u^n_{m+1}-2u^n_m+u^n_{m-1}$

Backward-time/central space approximation:

$$\frac{u^{n+1}_m-u^n_m}{k}=\frac{b}{h^2} \delta^2_x u^{n+1}_m+f^n_m$$

Exercise

Show that the above scheme is unconditionally stable of order $(1,2)$, and dissipative when $$\mu\equiv \frac{k}{h^2}$$ is bounded away from 0.

Crank-Nicholson (CN)

An old-time favorite:

$$\frac{V^{n+1}_m-u^n_m}{k}=\frac{1}{2 h^2}b\left(\delta^2_x u^{n+1}_{m}+\delta^2_x u^n_m\right)+\frac12\left(f^{n+1}_m+f^n_m\right)$$

Exercise

Show that CN is implicit, unconditionally stable, of order $(2,2)$. Furthermore, show that it is dissipative of order 2 if $\mu$ is constant, but not dissipative if $\lambda=\displaystyle \frac{k}{h}$ constant.