Convergence of numerical method for Burgers' equation

Question

This is a classic question from Leveque's book Numerical Methods for Conservation Laws (Exercise 12.4 p. 135).

Consider the 1-dimensional Burgers equation $u_{t}+(\frac{u^{2}}{2})_{x}=0$ with the inital data: $$ u(x,0)= \begin{cases} -1, & x < 1\\ +1, & x>1\\ \end{cases} $$ Let $k=\frac{1}{2l}$, $h=\frac{1}{l}$, where $l$ is a positive interger, $k$ is the step length in $t$-direction, $l$ is the step length in $x$-direction. We can discretize the inital data as: $$ u_{j}^{0} = \begin{cases} -1, & j<l \\ 0, & j=l\\ 1, & j>l \end{cases} $$ Consider the upwind method $u_{j}^{n+1}=u_{j}^{n}-\frac{k}{h}(F(u_{j}^{n},u_{j+1}^{n})-F(u_{j-1}^{n},u_{j}^{n}))$ with flux: $$ F(v,w)= \begin{cases} f(v), & \frac{f(v)-f(w)}{v-w}\geq 0 \\ f(w), & \frac{f(v)-f(w)}{v-w}< 0 \\ \end{cases} $$ We need to prove the sequence $u_{j}^{n}$ converges to the rarefaction wave solution as $l \rightarrow \infty$.

The (transonic) rarefaction wave solution is: $$u(x,t)= \begin{cases} {-1}, & x<1-t \\ \frac{x-1}{t}, & 1-t<x<1+t \\ 1, & x>1+t \end{cases} $$ I have spent a lots of days in this question but still I don't know how to do. Can anyone help me?

Answer 1

Apologies if I can't provide a hammer proof for this, but I don't know what kind of argument LeVeque is expecting. Still, there are a few things that can be mentionned. Let's begin with the following remarks:

• Here, $f(u) = \frac12 u^2$ and therefore $\frac{f(v)-f(w)}{v-w} = \frac12 (v+w)$. If the vector $U_j^n = (u_{j-1}^n, u_j^n, u_{j+1}^n)$ with values in $[-1,1]$ has constant sign and is non-decreasing w.r.t. $j$, then according to the upwind scheme, we can provide the bounds $$ u_j^{n+1} \in \left\lbrace \begin{aligned} & \big[u_j^n - \tfrac14 (u_j^n)^2, u_j^n\big] \subset [0,1] \quad\text{if}\quad U_j^n \in [0,1]^3_\nearrow\\ & \big[u_j^n, u_j^n + \tfrac14 (u_j^n)^2\big] \subset [-1,0[ \quad\text{if}\quad U_j^n \in [-1,0[^3_\nearrow \end{aligned}\right. $$ Therefore, provided $U_j^n$ remains non-decreasing w.r.t. $j$, we know that the sequence $0\leq u_j^{n} \leq 1$ is decreasing in time and that the sequence $-1\leq u_j^{n} < 0$ is increasing in time, while both sequences remain inside their bounds. Lastly, if $u_j^n = 0$ with $u_{j-1}^n < 0$ and $u_{j+1}^n \geq 0$, then $u_j^{n+1} = u_j^n$ vanishes too. Hence, everything seems to indicate that we would get close to the centered (transonic) rarefaction wave already after the first iteration in time.

Then I would probably (1) look at the stability properties of the scheme (monotonicity). In fact, the scheme is conservative and consistent with the PDE, hence the Lax-Wendroff theorem tells us that it will necessarily converge towards a weak solution (if only it converges). To know whether we will converge to the correct one, we need the notion of mathematical entropy. For the Burgers equation, $u^2$ is a mathematical entropy. Then, I would try to (2a) expand the time-stepping formula for $u^2$ or (2b) try to estimate the numerical viscosity.