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.
Numerical methods for the $u_t + \frac{(u_x)^2}{2} = 0$ equation
hyperbolic pde;oscillations
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.