Methods for Hyperbolic Equations

The FTCS-Method can also be applied on Hyperbolic equations of the form $u_{tt}=u_{xx}$. By using the approximation for the second derivatives (31) for both sides of the equations we obtain:

$$\frac{u_l^{m-1}-2u_l^m+u_{l}^{m+1}}{k^2} = \frac{u_{l-1}^m-2u_l^m+u_{l+1}^m}{h^2}$$

$$u_l^{m+1} = \frac{k^2}{h^2}(u_{l-1}^m-2u_l^m+u_{l+1}^m) +2u_l^m-u_l^{m-1}$$ (35)

To become a stable approximation, the constraint $${0<\frac{k^2}{h^2}\le1}$$ has to be hold. By setting $${r=\frac{k^2}{h^2}=1}$$, (35) becomes:

$$u_l^{m+1} = u_{l-1}^m+u_{l+1}^m-u_l^{m-1}.$$ (36)