# High-order upwind and non-oscillatory approach for steady state diffusion, advection-diffusion and application to magnetized electrons

## TL;DR

This paper develops high-order upwind and WENO schemes for steady state magnetized electron fluid equations, demonstrating their accuracy and ability to capture sharp gradients without oscillations, especially under strong magnetic confinement.

## Contribution

The paper introduces a novel application of higher order upwind and WENO schemes to magnetized electron fluid equations and advection-diffusion problems, showing their effectiveness in capturing sharp gradients.

## Key findings

- Linear schemes suffice for diffusion equations.
- WENO schemes do not cause adverse effects on solutions.
- WENO captures sharp gradients effectively in magnetized electron simulations.

## Abstract

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.