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I would like to collect some test-problems for nonlinear hyperbolic PDEs (Euler Equations, Shallow Water Equations, Ideal MHD, Acoustic Perturbation, ...) for which analytical solutions are known.

A famous example of this is the isentropic vortex advection by Shu for the 2D Euler equations. For the Euler equations, some more examples are also given in Chapter 7.13 of this book. In 1D, often also solutions (at least for Riemann problems) can be constructed. Examples of this include Burger's Equation and the LWR traffic model.

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    $\begingroup$ Might be better suited for Mathoverflow than Computational Science $\endgroup$
    – user9794
    Commented Nov 17, 2022 at 9:43
  • $\begingroup$ Probably worth a try, I could also give Physics SE a shot. $\endgroup$
    – Dan Doe
    Commented Nov 17, 2022 at 10:59

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A large and important class of problems with known exact solutions is given by the 1D Riemann problem, consisting of piecewise constant initial data with two initial states. For instance, this book (which I co-authored) provides many examples of exact solutions of Riemann problems.

More generally (and especially if you want smooth solutions, or solutions in multiple dimensions) you can use the method of manufactured solutions. In this approach you simply choose the exact solution, substitute it into the PDE, and find out what boundary and initial conditions are required.

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Take a look at the Wikipedia page listing non-linear PDEs. Quite a few of them have exact solutions (though they are not all hyperbolic).

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