5
$\begingroup$

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:

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

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.

$\endgroup$
  • 2
    $\begingroup$ The general solution is given implicitly: $u = f(x+u^2t)$. The actual tricky tricky part comes from applying the initial conditions. $\endgroup$ – Dylan Mar 27 at 8:59
  • $\begingroup$ Treat it as a constant. $\endgroup$ – dmtri Mar 27 at 9:01
  • 2
    $\begingroup$ @BAYMAX You may refer to this answer on how to deal with discontinuities $\endgroup$ – Dylan Mar 27 at 9:06
  • 2
    $\begingroup$ This similar problem may also be of interest. $\endgroup$ – Dylan Mar 27 at 9:25
  • $\begingroup$ @Dylan Yes indeed, but nonconvex conservation laws (such as the present one - and not the linked posts) need a careful treatment. $\endgroup$ – Harry49 May 23 at 9:43
5
$\begingroup$

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:

characteristics

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):

hull

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.