Convergence of finite difference scheme for conservation law I am currently studying for a test and came across this problem. The PDE is given by $$u_t + f(u)_x = 0 $$
where $f(u) = u^2/2$ , $u$ is periodic with period $2 \pi$ in $x$, and $u(x,0) =u_0(x)$ is  a given smooth periodic function. The problem gives us the two step finite different scheme \begin{align*}
\hat{u}_j^{n+1} &= u_j^n - \lambda (f(u_{j+1}^n)-f(u_j^n)) \\
u_{j}^{n+1} &= \frac{1}{2}(u_j^n + \hat{u}_j^{n+1}) - \frac{\lambda}{2}(f(\hat{u}_{j}^{n+1})-f(\hat{u}_{j-1}^{n+1}))
\end{align*}
and asks us to find a constraint on $\lambda = \frac{\Delta t}{\Delta x}$ so that the scheme converges as $\Delta x, \Delta t \to 0$. According to chapter 12 in this text, a method will converge to the weak solution if it is consistent with the conservation law, the numerical flux function is Lipschitz continuous, and the method is $TV$ stable. Consistency should just follow by doing taylor expansions. I determined the numerical flux function to be $F(u_j^n,u_j^{n+1}) = \frac{f(\hat{u}_{j}^{n+1}) + f(u_{j+1}^n)}{2}$, which is Lipschitz in its arguments. I am not so sure how to prove TV-stability though. In the pdf, the author generally will prove TV-stability by first proving that the method is $L_1$ contracting, i.e $$\|u^{n+1} \|_1 \leq \| u^{n} \|_1,  $$
but I've had no luck showing this. Am I going about this the right way? Any help would be very much appreciated.
 A: This is the MacCormack method (see also this Wikipedia article, Sec. Jacobian free methods) for the inviscid Burgers equation. Combining both steps, we can rewrite the method as
$$
u_j^{n+1} = u_j^n - \frac{\lambda}{2} \big(f(u_{j+1}^n) + f(\hat u_j^{n+1}) - f(u_{j}^n) - f(\hat u_{j-1}^{n+1}) \big)
$$
with $\hat u_j^{n+1} = u_j^n - \lambda\big(f(u_{j+1}^n) - f(u_j^n)\big)$.
Thus, the flux function proposed in OP is correct, and the scheme is consistent. It remains to prove stability.
Instead of proving the $L_1$-contraction property (as proposed in OP), let us prove the $l_1$-contraction property (see [1], p. 167-168). For this purpose, let's introduce $w_j^n = u_j^n - v_j^n$ with compact support. We introduce also $\hat w_j^{n+1} = \hat u_j^{n+1} - \hat v_j^{n+1}$, which is given by
$$
\hat w_j^{n+1} = w_j^n - \lambda \big(f(u_{j+1}^n) - f(v_{j+1}^n) - f(u_{j}^n) + f(v_{j}^n)\big) .
$$
In what follows, we'll make use of the fact that $f(b)-f(a) = \frac{a+b}{2}(b-a)$. By using the scheme's formula above, we rewrite $w_j^{n+1} = u_j^{n+1} - v_j^{n+1}$ as
$$
w_j^{n+1} = \left(1 + \frac{\lambda}{2}\tfrac{u_{j}^n + v_{j}^n}{2} \right)w_j^n - \frac{\lambda}{2} \left(\tfrac{u_{j+1}^n + v_{j+1}^n}{2}w_{j+1}^n + \tfrac{\hat u_{j}^{n+1} + \hat v_{j}^{n+1}}{2}\hat w_{j}^{n+1} - \tfrac{\hat u_{j-1}^{n+1} + \hat v_{j-1}^{n+1}}{2}\hat w_{j-1}^{n+1} \right)
$$
with
$\displaystyle
\hat w_j^{n+1} = \left(1+\lambda\tfrac{u_{j}^n + v_{j}^n}{2}\right)w_j^n - \lambda \tfrac{u_{j+1}^n + v_{j+1}^n}{2}w_{j+1}^n
$. To complete the proof, it remains to prove that $\|w^{n+1}\|_1 \leqslant \|w^{n}\|_1$.
Alternatively, it may be simpler to try proving that $$\|w^{n+1}\|_1\leqslant \|\hat w^{n+1}\|_1 \leqslant \|w^{n}\|_1$$ under a given CFL restriction. This property would then imply $l_1$-contraction and thus TV-stability.
[1] R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser, 1992.
