.. _sphx_glr_api_gallery_pde_conservation_laws:

Conservation laws and shocks
----------------------------

Inviscid Burgers' equation: shocks, rarefactions, and Godunov's
finite-volume method against oscillating second-order schemes.


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    <div class="sphx-glr-thumbcontainer" tooltip="In inviscid Burgers&#x27; equation u_t + (u^2/2)_x = 0, faster fluid overtakes slower fluid and smooth data steepens into a shock. Sergei Godunov (1959) computed the flux between neighboring cells from the exact solution of the Riemann problem they pose. The result is a conservative, monotone scheme that moves shocks at the right speed without spurious wiggles. He also proved that no linear scheme can be both monotone and better than first order, which is why second-order Lax-Wendroff oscillates. This script shows a sine wave breaking into a shock, then compares three schemes on a Riemann problem.">

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  :doc:`/api/gallery/pde/conservation_laws/plot_01_godunov_shocks`

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      <div class="sphx-glr-thumbnail-title">Godunov's method: capturing shocks without oscillations</div>
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    </div>


.. toctree::
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   /api/gallery/pde/conservation_laws/plot_01_godunov_shocks