📂Partial Differential Equations

Solution to the Riemann Problem for the Burgers' Equation

Description

$$\begin{cases} u_{t} + u u_{x} = 0 & , t>0 \\ u(t,x) = \begin{cases} a & ,x<0 \\ b & ,x>0 \end{cases} & , t=0 \end{cases}$$

The Riemann problem refers to the case where the solution to the Burgers’ equation, given an initial value, is expressed as a step function. In this case, if we have $a \ne b$, the obtained solution would have multiple or no functional values in certain intervals. Thus, the consistency rule is applied, or a smoothed solution is obtained.

These solutions satisfy the Rankine-Hugoniot condition and the entropy condition.

Solution

• Case 1. $a>b$

  

  The wave breaks as shown above.

  Thus, applying the consistency rule, we obtain the solution

  $$u(t,x) = \begin{cases} a &, x < {{a+b} \over {2}} t \\ b &, x > {{a+b} \over {2}} t \end{cases}$$.

• Case 2. $b>a$

  

  The wave, as shown above, requires smoothing to assign a functional value in the interval where it does not exist.

  Therefore, we obtain the solution

  $$u(t,x) = \begin{cases} a &, x < a t \\ x/t & , at \le x \le bt \\ b &, x \ge b t \end{cases}$$.