Behavior of the solution to the inviscid Burgers’ equation

characteristicshyperbolic-equationspartial differential equations

Consider the inviscid Burgers' equation $u_t+uu_x=0$ with the initial condition $$u_0=\begin{cases}
0, & x<0\\
x, & 0\leq x \leq 1\\
1, & x>1
\end{cases}$$

I tried to implement numerical method to solve this and I got this solution:enter image description here

Can somebody pls explain the behavior of this solution? I got the following characteristics:enter image description here

Best Answer

The method of characteristics gives the solution $u = u_0(x-ut)$, where $u_0$ is the initial data. In the present case, we have

  • $u=0$ if $x-0\cdot t\leq 0$;

  • $u=1$ if $x-1\cdot t\geq 1$;

  • $u=x-ut$ i.e. $u=x/(1+t)$ otherwise;

so that $$ u(x,t) = \left\lbrace \begin{aligned} &0 & & \text{if}\quad x \leq 0 ,\\ &\tfrac{x}{1+t} & & \text{if}\quad 0 \leq x \leq 1+ t ,\\ &1 & & \text{if}\quad 1+ t \leq x . \end{aligned} \right. $$ The previous analysis seems consistent with the numerical results in OP, up to the right boundary $x=3$ where a Dirichlet boundary condition may have been implemented. An outflow boundary condition would be more appropriate.

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