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I'm looking for some methods that could be directly applied to the PDE $$ \frac{\partial u}{\partial t} + \frac{(u_x)^2}{2} = 0\tag{*} $$ without converting it by $v = u_x$ to the Hopf equation $$ \frac{\partial v}{\partial t} + \frac{\partial (v^2/2)}{\partial x} = 0. $$

Ideally, a WENO scheme is perfect, but I'm wondering how to apply a limiter or a numerical flux to $(*)$.

Simple centered scheme $$ \dot u_i(t) = -\frac{1}{2}\left(\frac{u_{i+1}(t) - u_{i-1}(t)}{2h}\right)^2 $$ expectedly oscillates when shock (on $v = u_x$) is formed. enter image description here

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  • $\begingroup$ You have tagged this with "hyperbolic-pde", but it is really a Hamilton-Jacobi equation. Have you read anything about discretization of Hamilton-Jacobi equations? $\endgroup$ – David Ketcheson Oct 18 '15 at 9:45
  • $\begingroup$ @DavidKetcheson Thank you David, I'm going to read about HJ equation now. $\endgroup$ – uranix Oct 18 '15 at 9:47
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This is a Hamilton-Jacobi equation. You can read about how to apply WENO to such equations in Section 4 of Chi-Wang Shu's 2009 WENO review paper, and references therein.

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The simplest scheme you could employ is an upwind scheme. It's first-order and introduces artificial viscosity/diffusion, but doesn't require limiters. Probably the next simplest class would be Lax-Wendroff schemes, for which you can find a comprehensive explanation in LeVeque's book on Finite Volume schemes.

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  • $\begingroup$ Can you please show me the upwind scheme for my equation? Not for the Hopf form, but for the original one. $\endgroup$ – uranix Oct 14 '15 at 18:54

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