I want to solve the PDE:

$$\frac{\partial u}{\partial t}+(1-2u)\frac{\partial u}{\partial x}=0 $$

for $x \leq0 $ and $t \geq 0$, with initial condition $u(0,x)=\frac{1}{2}$, and boundary condition;

$u(t,0)=\left\{\begin{matrix} \frac{t}{4}+\frac{1}{2}, 0\leq t\leq 2 & \\ 1, t \geq 2& \end{matrix}\right.$

by using the method of characteristics.

The method of characteristics gives us that $u$ is constant along characteristics which have equation defined by $\frac{dx}{dt}=1-2u$.

For a characteristic originating from the x axis at a point $(x,t)=(x_0,0)$, this tells us that on characteristics defined by $x(t)=x_0$, we have $u(x(t),t)=u(x(0),0)=\frac{1}{2}$. i.e. the characteristics originating on the x axis are vertical straight lines on which $u=\frac{1}{2}$.

For a characteristic originating from the t axis at a point $(x,t)=(0,\tau)$, we have 2 cases, depending on the value of $\tau$.

In both cases $\frac{du}{dt}=0$, on $\frac{dx}{dt}=1-2u$, so u is constant on the characteristics, so $u(x(\tau),\tau)=u(0,\tau)$ on characteristics defined by $x(t)=(1-2u(0,\tau))t+C$, where $C$ is constant. Applying our boundary condition this gives us:

$u=\frac{\tau}{4}+\frac{1}{2}$, on $x(t)=-\frac{\tau}{2}(t-\tau)$, for $0\leq\tau\leq2$, and:

$u=1$ on $x(t)=-(t-\tau)$, for $\tau\geq2$.

Up to here I'm fine, but when I draw up a plot of the characteristics in the t-x plane I get very confused. The x-axis characteristics are all vertical lines. The t-axis characteristics for $0\leq\tau\leq2$ are straight line originating from $(x,t)=(0,\tau)$ with slope that varies. As $\tau$, decreases from 2 to 0, the slope of these characteristics (in the t-x plane), goes from -1 and approaces $-\infty$, as $\tau$, approaches zero, meaning that each of these characteristics intersect.

As $u$ is constant on a characteristic, this is a contradiction, and so we expect a shock wave (implying a discontinuous solution).

Also for $\tau\geq2$, the characteristics are all parallel lines of slope -1, simply translated in the positive t direction (upwards in the t-x plane).

What is really confusing me is that the three types of characteristics are all intersecting, and that the $0\leq\tau\leq2$ characteristics are intersecting themselves infinitely often, and this is maling it hard for me to apply the shock criterion. At the origin, the characteristics firast intersect, so we expect a shock will immediately form. It will propagate with shock velocity given by: $\frac{dx_s}{dt}=\frac{(u(1-u))|_+-(u(1-u))|_-}{(u)|_+-(u)|_-}$

Where the subscripts denote the value of u just to the right or jsut to the left of the shock.

I really don't know how to continue from here, especially because of the three types of intersecting characteristics and the $0\leq\tau\leq2$ characteristics intersecting with themselves. I would really be grateful for any hints or ideas or help. I've tried a lot of things to resolve this issue, but none have gotten anywhere and it would take a long time to write them out. I think if I can just undbrstnad the shock condition/formation here, I can solve this problem.


An alternative way for solving thanks to the method of characteristics. So, you can compare with your result.

enter image description here enter image description here

The next figure shows $u$ as a function of $x$ at various time.

enter image description here


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