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Burgers Equation reads as $u_t + (\frac{u^2}{2})_x = 0 \quad (t>0,x\in R)$. I'm recently studying hyperbolic conservation law. My question is how can I compute the unique entropy solution with given initial data. Typically, how can I get the solution when the initial data is given by:

  1. $$ u_0(x) = \begin{cases} 0 , |x|\geq 1\\ 1,|x| <1\end{cases}$$
  2. $$ u_0(x) = \sin(x)+2$$

If we can't get the close formula, how can i get the precise entropy solution numerically? I know there are some finite differential scheme, but the solution might have dissipation or oscillation which I think is not very precise.

Are there any general method to calculate the entropy solution with given initial data?

Add: Thanks for comment. I think the R-H Condition can help me capture the behaviour when my solution has discontinuity. But I don't know what to do with the continuous case where characteristics get crossed.

EditPiAf
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EggTart
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1 Answers1

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The entropy solution is the same as the classical solution (obtained with the method of characteristics) as long as the latter keeps smooth. Otherwise, we need to ensure the admissibility of discontinuities by using the appropriate entropy condition.

For 1., we have already discontinuous initial data. More precisely, the data is piecewise constant, so that we need to solve Riemann problems. There are analytical solutions for this problem (cf. the many similar examples on this site).

For 2., the initial data is continuous. Thus, we can use the implicit solution $u = u_0(x-ut)$ deduced from characteristics until the breaking time. However, there is no known closed-form solution (cf. the many similar examples on this site). Then, the solution becomes discontinuous, and similar methods to 1. are employed.

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