I have a set of modified compressible Euler equations that I would like to solve using a WENO method. The issue is that the modified flux function involves derivative and filtering terms and I'm not sure how to compute the derivative terms in a consistent manner within the WENO scheme. For simplicity, the key aspects of the problem can be illustrated with the modified 1D Burgers equation$$u_t + \left( \dfrac{1}{2}u^2+\dfrac{3}{2}\alpha^2\overline{(u_x)^2} \right)_x = 0$$where the overbar denotes a filtering operation using the Helmholtz filter$\overline{f} = g^\alpha*f$and$$g^\alpha(x) = \dfrac{1}{2\alpha}e^{-|x/\alpha|}$$If $\alpha$ is on the scale of the grid spacing, this PDE produces near discontinuities at the scale of the grid spacing so I would like to use a WENO method for simulations.My question: given the values of $u$ at the grid points, how should I compute $u_x$ at the grid points so that I can reconstruct the flux function in a WENO fashion since there may be near-discontinuities in $u$.Alternatively, would it be easier/more sensible to reconstruct $u$ and $u_x$ at half-grid points and then compute the flux function from those reconstructed values? If so, how should I compute $u_x$ at the half grid points?
WENO reconstruction of flux involving derivative terms
fluid dynamics;hyperbolic pde