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Plasma Dynamics Group − Aerospace & Energetics Research Program − University of Washington

Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models

C. Aberle, A. Hakim, and U. Shumlak

Aerospace and AstronauticsUniversity of Washington, Seattle

American Physical Society - Division of Computational PhysicsAugust 2002

San Diego, CaliforniaFor a copy of this poster, visit http://www.aa.washington.edu/cfdlab


Abstract

A Maxwell Equation Solver is developed for use in two fluid plasma solvers. The application requires that the solver be adaptive to the geometry, handle discrete currents and charge distributions, and capture dynamic behavior. Two approaches, Finite Volume Time Domain (FVTD) and Mixed potentials, are investigated, compared, and results are presented. The FVTD scheme, Weighted Essentially Non-Oscillatory (WENO) is applied to Cartesian and triangular meshes. Solvers using Cartesian meshes perform adequately without currents, but they fail to accurately model situations with discrete currents. Solvers using triangular meshes successfully overcome this problem by reducing cross-term dissipation. The mixed potential solver is second order accurate in space and time. It uses quadratic upwind for the spatial fluxes and a central differencing for the time. Successful results are obtained in simulations with discrete currents.


Motivation

Many current plasma simulation codes are based on the magnetohydrodynamic (MHD) model whose derivation involves several assumptions that severely limit its applicability, particularly for Hall effect physics. The two-fluid plasma model only assumes local thermodynamic equilibrium for each species and, therefore, more accurately models the appropriate physical processes. Accurate electromagnetic (EM) field solvers are needed to properly account for the interactions between the fields. The solver must be capable of handling propagation of EM waves in all directions; and must be able to account for charges and currents as both continuous and discrete sources. In this work, we develop and compare two different approaches to developing an EM field solver. The first approach solves the Maxwell equations using Finite Volume Time Domain (FVTD) methods and the second approach uses a mixed potential formulation with a quadratic upwind differencing scheme.


Maxwell Equations

● These equations are solved in two dimensions.

● The transverse electric mode subset of these equations are used.


Conservative Form of Equations

Using a non-dimensionalization to set , Maxwell's equations are written in conservative form:

where


Potential Formulation

where

with and connected by the Lorentz gauge condition


Potential Formulation

These can be transformed from the physical space to the grid space using transformation formula for the Laplacian

for each and where is the metric tensor, and is divergence operator in space.


Discretization Schemes

To solve the Maxwell Equations, they must be discretized. Two approaches are used, Finite Volume Time Domain (FVTD) and potential formulation.

The FVTD approach divides the domain into cells, and tracks the movement of quantities between the cells. This approach is based upon the methods of computational fluid dynamics (CFD) of the past several decades. Shang calls these 'characteristic methods'.

The potential formulation uses a finite difference scheme to discretize the potential form of the Maxwell equations. The spatial derivatives are discretized using Quadratic Upwind Scheme (QUICK).


Finite Volume Time Domain (FVTD)

For FVTD, integrating the conservative form of Maxwell equations over a cell, we get

where is the cell area and is the length of the cell edge . is a numerical flux, the movement of the conserved variables between cells in the domain. Each FVTD scheme calculates and uses this value to update .

FVTD is tested on two different types of grids, one with rectangular shaped cells, and the other with triangular shaped cells.


Upwind with Finite Volume

The first order upwind finite volume scheme calculates the numerical flux using the quantities of the nearest cell upwind of the cell edge. For Maxwell equations,

where these and are to lie in the x direction.

Temporal discretization is accomplished using a simple first order Euler time integration.


Weighted Essentially Non-Oscillatory (WENO) Scheme

WENO schemes are a variation on Essentially Non-Oscillatory (ENO) schemes. WENO and ENO schemes were developed by Harten, Shu, and others over the past decade. ENO schemes interpolate the flux from neighboring cells. An ENO scheme chooses those neighboring cells that have the smoothest solution.

To form an ENO scheme, stencils are made of the cells neighboring the cell edge. For example, to interpolate on to the cell edge as shown, three stencils would be created for a 2nd order solver. The first stencil would use cell 1 and 2. The second stencil would use cell 1 and 3. The third stencil would use cell 1 and 4.For rectangular grid cells, these stencils are used on the axis normal to the cell edge.


WENO

WENO schemes recognize that for smooth solutions, all the cells can be safely used for interpolation. Therefore, instead of selecting the interpolation of only a single stencil, WENO performs a weighted average on the interpolations of every stencil. The stencils are weighted in such a way that a really smooth solution will be using a higher order interpolation consisting of all the cells.

Using WENO schemes with first order Euler time is unstable. Time integration methods of higher order are needed. Using WENO on a rectangular grid uses an explicit Runge Kutta method of 3rd order.

Using WENO on a rectangular grid requires the use of an implicit time method. The implicit time method uses a residual iterative technique for each time step. By applying conjugate gradient repeatedly, the residual is driven towards zero for the time step.


Potential Formulation: Temporal Discretization

To advance the solution in time, the temporal derivatives are discretized using central differencing scheme:


Potential Formulation: Spatial Discretization

The spatial derivative are discretized by fitting a quadratic between two upwind points and one downwind point

where are constants determined from the grid spacing and the values of at the upwind and downwind points. In practiceit is simplest to explicitly determine from values at the neighboring nodes and then use them in calculating the needed derivatives.

For uniform rectangular grids, the differencing formula reduces to


Sine Wave

This simulation tests the propagation of an initial sine wave across the domain with periodic boundary conditions. The simulation runs for one time unit, at which time, the solution should look exactly like the original waveform. This gives a good measure of dissipation.

Solid lines shows the exact solution while dotted lines shows the solution obtained with QUICK scheme.

This uses 100 points while the FVTD used 20 points.

Standing wave in 1-D: QUICK scheme

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-1

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0

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0.8

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Ex


Sine Wave

This shows the results of upwind on a rectangular mesh. As is shown, increasing the resolution of the mesh decreases the error, as anticipated.

This shows the results of using 2nd

order WENO on a triangular mesh. As is shown, increasing the resolution of the mesh decreases the error, as shown. Note, the accuracy for the 20x20 is much better than the rectangular upwind.


Square Wire

Solid lines shows the exact solution while dotted lines shows the solution obtained with QUICK scheme.

The schemes must be able to handle discrete currents within the domain. Therefore, a square wire is simulated and results are compared. This square wire is centered in the domain with absorbing boundary conditions. The simulation runs until steady state is achieved The resulting magnetic field is compared against the analytic magnetostatic solution. This analytic solution is determined by applying the Biot Savart law to this square wire. The size of the domain is ten times in length and a hundred time in area with respect to the wire.

Magnetic field around square Wire

0

0.5

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-0.18 0.02 0.22 0.42 0.62 0.82 1.02 1.22 1.42

Radial distance from wire center

Rad

ial m

agne

tic fi

eld


Square Wire, Success & Failure

Triangular Grid Rectangular Grid

By looking at these plots, the upwind on the rectangular grid is an obvious failure. The upwind on the triangular grid is generating a reasonable solution. Therefore, the upwind on the rectangular grid is unable to a solution that is qualitatively reasonable.


Square Wire, FVTDRectangular GridTriangular Grid

These plots illustrate the accuracy of the solvers. These plots are taken along the x=y diagonal of the domain. The upwind on the triangular grid performs quite, whereas the upwind on the rectangular grid reiterates its failures.


Analysis of Rectangular vs Triangular Grids

The behavior of FVTD on rectangular and triangular grids is analyzed using the Von Neumann method. This type of analysis looks at the behavior of Fourier components to the solution. While Von Neumann is typically used to determine stability, the method obtains information about dissipation. The dissipation of the schemes is crucial to explain their behavior in the simulation of the square wire.

Each Fourier component is designated as where the range is from 0 to +1. Each component is a separate 2D wave mode that has the spatial form

The Von Neumann method calculates the Gain G that each mode experiences for each time step. If |G|>1, then the scheme is unstable. If |G|=1, then there is no dissipation. If |G|<1, then the scheme is dissipative.This method is applied to both the rectangular and triangular upwind FVTD schemes.


Rectangular vs Triangular Grids for the Scalar Wave Equation


This shows the dissipation behavior of the simple scalar wave propagation in the +xy diagonal direction. As can be seen by both mesh plots, the dissipation is acceptable and both grids would perform adequately. Given the simplicity of a rectangular grid, it is more favorable for the scalar equation.


Rectangular vs Triangular Grids for the Transverse Electric Mode (TEM)


These plots show the dissipation for the electric field component of TEM. Yikes, the dissipative behavior of the rectangular grid for the TEM is vastly different than for the scalar wave equation. It is this massive dissipation that accounts for the inability for the rectangular upwind to propagate along a diagonal.The triangular upwind is capable of diagonal propagation and this is justified by this dissipation plot.


Summary

Electromagnetic field solvers for use in plasma simulations are developed in this work. Two different approaches are taken: a Finite Volume Time Domain (FVTD) approach and an approach using the mixed potential formulation.

The FVTD method gives accurate results for simple problems. For problems requiring sources, FVTD on rectangular grids fails while it succeeds on triangular grids.

The mixed potential formulation using quadratic upwinding givesaccurate results on arbitrary two dimensional grids for the testproblems considered. It is seen that the solver is able to account fordiscrete sources well. Although preliminary results look promising,much work needs to be done to test the efficacy of this approach inplasma simulations.


Summary

The two methods developed give comparable results, at least for thetest problems considered. The FVTD method is more flexible as it candeal with arbitrary grids, while the potential solver needs atransformation from physical to logical space. Further, as thepotential solver uses finite differences, it may be difficult tohandle shocks using it. Also, the conversion to E and B fields fromthe potentials involves a loss in accuracy.