Lax–Friedrichs method

Remarks

It is in conservation form $u_j^{n+1} = u_j^n - \lambda \left( G_{j+1/2} - G_{j-1/2} \right)$ with the numerical flux $$G_{j+1/2} = \frac{F(u_j^n) + F(u_{j+1}^n)}{2} - \frac{u_{j+1}^n - u_j^n}{2 \lambda}.$$ Its truncation error is of first order in $\Delta x$ when $\lambda$ is fixed.

Example: for $u_t + a u_x = 0$ with $a \in \R$, the scheme reads $u_j^{n+1} = \frac{1 - a\lambda}{2} u_{j+1}^n + \frac{1 + a\lambda}{2} u_{j-1}^n$. If $\abs{a} \lambda \le 1$, both weights lie in $[0, 1]$ and sum to $1$. So $\max_j \abs{u_j^{n+1}} \le \max_j \abs{u_j^n}$, and the scheme is stable in the maximum norm.

Non-example: the forward-time centred-space scheme $u_j^{n+1} = u_j^n - \frac{\lambda}{2}(F(u_{j+1}^n) - F(u_{j-1}^n))$ differs only in the first term. For $u_t + a u_x = 0$ with $a \ne 0$, it multiplies the Fourier mode $e^{i j \xi}$ by $1 - i a \lambda \sin \xi$, of modulus greater than $1$ when $\sin \xi \ne 0$. So it is unstable for every $\lambda > 0$.