Solving the Quasilinear PDE $u_{t} - u^2 u_{x} = 0$ with piecewise initial condition.

Solving the quaslinear PDE by the method of characteristics is a bit tricky for me.

I was trying to obtain the solution $$u$$ for the PDE

$$u_{t} - u^2 u_{x} = 0$$ The initial condition is given by:

$$u(x,0) = g(x) = \begin{cases} -0.5, & x \leq 0 \\ 1, & 0

Using the method of characteristics I have tried to write the characteristic equation as

$$\frac{dt}{1} = \frac{dx}{-u^2}$$

Trying to obtain an expression for $$u$$ from the above equation seems tricky as there is no $$u$$ term.

• The general solution is given implicitly: $u = f(x+u^2t)$. The actual tricky tricky part comes from applying the initial conditions. – Dylan Mar 27 at 8:59
• Treat it as a constant. – dmtri Mar 27 at 9:01
• @BAYMAX You may refer to this answer on how to deal with discontinuities – Dylan Mar 27 at 9:06
• This similar problem may also be of interest. – Dylan Mar 27 at 9:25
• @Dylan Yes indeed, but nonconvex conservation laws (such as the present one - and not the linked posts) need a careful treatment. – Harry49 May 23 at 9:43

It is important to note that the flux $$f(u)=-\frac13 u^3$$ in the conservation law $$u_t + f(u)_x = 0$$ is nonconvex. The problem may be viewed as two neighbor Riemann problems, for which one can compute the waves, and their potential interactions. A plot of the base characteristic lines in the $$x$$-$$t$$ plane is given below: One observes that the method of characteristics provides a unique solution $$u = g(x + u^2 t)$$ in some specific parts of the plane only. In facts, there may be zero, one or two characteristics passing at a given point $$(x,t)$$. The previous plot suggest that a shock wave is generated at $$x=0$$, and that a rarefaction wave is generated at $$x=1$$. This claim is verified by convex hull constructions, a graphical method related to the Oleinik entropy condition for shock waves (see this post): Here the Rankine-Hugoniot condition gives the shock speed $$s = -\frac{1}{4}$$. Hence, the following solution for times $$t< \frac{4}{3}$$ is obtained: u(x,t) = \left\lbrace \begin{aligned} &{-\tfrac12} & &\text{if}\quad x< {-\tfrac{1}{4} t}\\ &1 & &\text{if}\quad {-\tfrac{1}{4} t} < x \leq 1-t\\ &\sqrt{(1-x)/t} & &\text{if}\quad 1- t \leq x \leq 1-\tfrac14 t\\ &{\tfrac12} & &\text{if}\quad 1- \tfrac14 t \leq x \end{aligned} \right. For larger times, the interaction between the shock and the rarefaction must be computed.