# WENO5 scheme in a staggered grid

I'm trying to use the finite-difference WENO scheme to solve the 2D density conservation law with axial symmetry (coordinates $$r,z$$):

$$\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v}) = \frac{\partial \rho}{\partial t}+ \frac{1}{r}\partial_r (r \rho v_r)+\partial_z (\rho v_z) = 0$$.

$$\rho$$ is defined at grid points $$i,j$$, $$v_r$$ at $$i+1/2,j$$ and $$v_z$$ at $$i,j+1/2$$.

Ideally, I would like use a WENO reconstruction procedure to estimate $$f_r = r \rho v_r$$ at $$i\pm 1/2,j$$, and $$f_z = \rho v_z$$ $$i,j \pm 1/2$$, from their values at $$i,j$$ (flux-splitted), and use those estimates to update the density at $$i,j$$:

$$\frac{\partial \rho^{i,j}}{\partial t} = -\frac{1}{r}\frac{1}{dr} \left[f_r^{i+1/2,j}-f_r^{i-1/2,j} \right]-\frac{1}{dz} \left[f_z^{i,j+1/2}-f_z^{i,j-1/2} \right]$$

Now, since the velocities aren't defined at $$i,j$$, I use a simple interpolation (i.e, $$v_r^{i,j} = (v_r^{i+1/2,j}+v_r^{i-1/2,j})/2$$) to estimate their values at those points.

Doing this, however, does not yield good results. I've tried to do a WENO interpolation of the densities, but that also isn't working as well as I need it to.

Any suggestions on how to handle this issue?

Additionaly, any recomendations on how to properly flux-split in this case? Most flux-splitting procedures seem to reduce to a trivial case when $$f(\rho) \propto \rho$$.

• I have not seen WENO being applied on staggered grids. Your choice of averaging the velocity would lose accuracy. You already have velocity at the faces (half indices). You could reconstruct density to the faces and then use a numerical flux. But may I ask why you are using staggered grids ? – cfdlab Jan 27 '19 at 4:02
• Thanks for the reply! I have reconstructed the densities as well, but also got results that weren't that good. I've been using Lax-Friederich flux in that case, since, say, Godunov flux just reduces to $\rho^{-,i+1/2,j} v^{i+1/2,j}$. Is there another flux you'd recommend? I've using a staggered grid, because this density update equation is coupled to Maxwell's equations, that do work well in a staggered grid. Is there another high order finite difference scheme you'd recommend? – grizzlyjoker Jan 27 '19 at 14:42