# Numerical methods for the $u_t + \frac{(u_x)^2}{2} = 0$ equation

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.

• 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? – David Ketcheson Oct 18 '15 at 9:45
• @DavidKetcheson Thank you David, I'm going to read about HJ equation now. – uranix Oct 18 '15 at 9:47