Development of a Maxwell Equation Solver for Application to Two

Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models C. Aberle, A. Hakim, and U. Shumlak Aerospace and Astronautics University of Washington, Seattle American Physical Society - Division of Computational Physics August 2002 San Diego, California For a copy of this poster, visit http://www.aa.washington.edu/cfdlab Plasma Dynamics Group − Aerospace & Energetics Research Program − University of Washington

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.

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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.

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

Maxwell Equations

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These equations are solved in two dimensions.

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The transverse electric mode subset of these equations are used.

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Conservative Form of Equations Using a non-dimensionalization to set in conservative form:

, Maxwell's equations are written

where

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Potential Formulation

where

with

and

connected by the Lorentz gauge condition

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Potential Formulation These can be transformed from the physical using transformation formula for the Laplacian

for each is divergence operator in

and where space.

space to the grid

is the metric tensor,

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

space

and

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).

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

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.

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

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

Weighted Essentially NonOscillatory (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.

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

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.

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

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

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

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. Standing wave in 1-D: QUICK scheme

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

1.2 1 0.8 0.6 0.4

Ex

0.2 0 0

0.1

0.2

0.3

0.4

0.5

-0.2

0.6

0.7

0.8

0.9

1

This uses 100 points while the FVTD used 20 points.

-0.4 -0.6 -0.8 -1 -1.2

x

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

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

Magnetic field around square Wire

2

Radial magnetic field

1.5

1

0.5

0 -0.18

0.02

0.22

0.42

0.62

0.82

1.02

1.22

1.42

Radial distance from wire center

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

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

Square Wire, FVTD Triangular Grid

Rectangular 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.

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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|