  Steady state simulations} of magnetized electron fluid equations with strong
anisotropic diffusion based on the first-order hyperbolic approach is carried
out using cell-centered higher order upwind schemes, linear and weighted
essentially non-oscillatory (WENO). Along with the magnetized electrons, the
diffusion equation is also simulated to demonstrate the implementation and
design order of the accuracy of the approach due to their similar upwind
structure. We show the adequacy of linear upwind schemes for diffusion equation
and the use of shock-capturing scheme like WENO does not have any adverse
effect on the solution, unlike the total-variation diminishing (TVD) methods.
We further extended the approach to advection-diffusion equation, and
appropriate boundary conditions have obtained a consistent design accuracy of
the third and fifth order. We implemented the WENO approach to
advection-diffusion equation by using the split hyperbolic method to
demonstrate the advantage of non-oscillatory schemes to capture sharp gradients
in boundary layer type problems without spurious oscillations. Finally,
numerical results for magnetized electrons simulations indicate that with
increasing strength of magnetic confinement it is possible to capture sharp
gradients without oscillations by WENO scheme.
