A Positivity Preserving Discontinuous Galerkin Finite Element Scheme for Modeling Electrical Discharges
Since the 1950's, interest in studying electrical discharges has been prevalent in the aerospace community, with applications that include electromagnetic flow control of high-altitude re-entry vehicles and design of plasma actuators. The Drift-Diffusion model is a popular discharge model, often used to simulate phenomena such as the glow and pulsed discharges. Currently used numerical solution strategies are mostly finite-difference schemes such as the Scharfetter-Gummel scheme, that are known to be highly dissipative. An attempt is made here to use high-order accurate DG methods to solve the Drift-Diffusion model. The popularity of DG methods has been increasing due to their capability of producing high-order accurate solutions on unstructured grids and the ease with which they can be parallelized. The main drawback, from an engineering point of view, is the high computational cost associated. It appears that DG methods have not yet been explored for solving the Drift-Diffusion model and hence, such an attempt is made here. We present a positivity-preserving DG scheme for the Drift-Diffusion model. The model consists of convection-diffusion equations solved concurrently with Gauss' law to predict discharge structures. A new DG scheme, that is positivity-preserving capable for convection-diffusion equations, is developed and applied to solve the plasma model. The new scheme can be viewed as a modification of existing DG methods and hence can be used adaptively/concurrently with formulations such as the Local Discontinuous Galerkin scheme. In view of this, an LDG framework for the Drift-Diffusion model is developed first, with steady state glow discharges in hydrogen gas as the main test case. The final product of this work involves an adaptation of positivity scheme along with the LDG formulation, employed to solve the Drift-Diffusion model, which is shown to predict accurate results for both transient and steady-state discharge phenomena. Scope for future work include a robust extension to higher-dimensional domains and studies that look into the capability of the scheme to predict discharges in other gases, as well as a mixture of species.
Zhang, Purdue University.
Applied Mathematics|Aerospace engineering|Plasma physics
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