Crank-Nicolson method

The constraint $${r=\frac{k}{h^2}\le\frac{1}{2}}$$ leads to very small time steps.
The difference quotient on the right hand side in (32) can be replaced by $\frac{1}{2}$ times the sum of two such difference quotients at two time rows to enable a larger step size in space and time direction:

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

$$+ \frac{1}{2h^2}(u_{l-1}^{m+1}-2u_l^{m+1}+u_{l+1}^{m+1})$$ (34)