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.
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.