a finite volume scheme for the two fluid plasma system

Aerospace & Energetics Research Program - University of WashingtonPlasma Dynamics Group

A Finite Volume Scheme for the Two Fluid Plasma System

J. Loverich, U. ShumlakAerospace and Energetics Research Program,

University of Washington

ICOPS26 - 30 May 2002


Abstract

A Finite Volume Scheme for the Two Fluid Plasma SystemJ. Loverich, U. Shumlak

University of Washington

Aerospace and Energetics Research Program Box 352250

Seattle, WA 98195-2250 In this paper we present our work on a numerical one-dimensional two fluid plasma solver. We take the collisionless, non-relativistic system of equations consisting of ion continuity, ion momentum, ion energy, combinedwith electron continuity, electron momentum and electron energy, coupled with the full electrodynamic Maxwell’s equations. The algorithm is a first-order time, second-order space finite volume formulation of a Roe-type approximate Riemann solver and uses flux limiters for good shock resolution without spurious oscillations. We address the issue of stiffness introduced by the speed of light and the stiffness associated with the strong coupling of the source terms in the hyperbolic system. Both explicit and implicit schemes are developed and the advantages of the implicit scheme are discussed. It is shown how the algorithm may be extended to multiple dimensions. The algorithm is tested on various numerical and physical problems including electrostatic and electromagnetic two fluid plasma waves and shock problems comparing the two fluid results to the MHD results.


Introduction

Numerical methods for solving the MHD equations have been studied extensively[1]. This system of equations is used to model experimental fusion devices, space propulsion concepts, space plasma physics and other problems involving plasma physics. The MHD equations, however, do not model potentially important plasma effects like space charge distributions, finite Debye length and finite gyro radius effects, electron inertia, ion current or grad-B drifts. Furthermore, numerical solutions involving the Hall term using the MHD equations are difficult. The MHD system of equations is derived from the two-fluid system and by solving the two fluid system all those effects just described can be modeled. Our technique is based off of work in computational fluid dynamics[3][5] and computational electromagnetics[4]. Ultimately we would like to be able to model a wide regime spanning the range from gas dynamic to weakly ionized gas to an MHD-like fluid, in its current incarnation the solver solves the full Maxwell’s equations including displacement current and is therefore able to model electromagnetic wave interactions with plasmas as well.


Fluid EquationsFor our fluid we take the inviscid, non-relativistic equations of fluid dynamics with the addition of the Lorentz force. Two species are used, electrons and ions, both of which are described by their own set of fluid equations. The two fluid equations allow for both electron and ion temperatures as well as electron and ion inertia. The fluid equations are written in normalized conservative divergence form as follows.

1. Species Continuity

2. Species Momentum

3. Species Energy ( ) αα

ααρααα

α UEm

qPeU

t

e⋅=+⋅∇+

∂

∂

121

−+⋅= γααααραpUUe

( )BUEm

qPUU

t

U×+⋅=∇+⋅∇+

∂

∂α

α

ααραααρααρ

0=⋅∇+∂

∂ααραρU

t

4.1=γ


Maxwell’s Equations

The full Maxwell’s equations are solved including displacement current. The Poisson constraint is satisfied initially and the Magnetic flux constraint is satisfied automatically because the problem is one dimensional. Faraday’s law and the Ampere’s law are used to update the electric and magnetic fields. Both electron and ion currents contribute to Ampere’s law. The equations are normalized such that the speed of light is 1.

1. Ampere’s Law

2. Faraday’s Law

3. Poisson’s Equation

4. Magnetic Flux

iUimiqi

eUemeqeB

tE ρρ

−−=×∇−∂∂

0=×∇+∂∂ EtB

imiqi

emeqeE

ρρ+=⋅∇

0=⋅∇ B


One Dimensional Form

The two fluid equations are solved in one dimension and Riemann solvers require the solution to one dimensional problems across each cell interface. In the following the one dimensional conservation form of the two fluid system is written.

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−−

−−

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−∂∂+

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂

000

0

0

izi

iiez

e

ee

iyi

iiey

e

ee

ixi

iiex

e

ee

y

z

y

z

z

y

x

z

y

x

UmqU

mq

UmqU

mq

UmqU

mq

EE

BB

x

BBBEEE

tρρ

ρρ

ρρIn one dimension Bx does not change with time, however, it is left as a conserved variable so that the algorithm can be more easily generalized to 3 dimensions.

One dimensional Maxwell’s equations

( )ααα

ρ

ρρ

ρρ

ρρρ

ρ

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

++−++−−+

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+

∂∂+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂

zzyyxx

xyyxz

xzzxy

yzzyx

x

zx

yx

x

x

z

y

x

EUEUEUBUBUEBUBUEBUBUE

mq

peUUUUUpU

U

x

eUUU

t

02

Each of the fluid equations has 4 source terms which are algebraic functions of the conserved variables of the fluid equations and of Maxwell’s equations.

One dimensional fluid equations


Solver

An Approximate Riemann Solver is used to solve each of the 3 systems of equations, electron fluid, ion fluid and Maxwell’s equations. The domain is broken into several finite volumes or cells and a Riemann problem is solved across each cell interface. The method chosen follows closely that outlined by Harten[2] and is formulated in the manner described by LeVeque[3]. The method is first order accurate in time and second order accurate in space except at shocks where the algorithm reduces to first order accurate in space. Flux limiters are used so that no spurious oscillations occur near shocks. A Roe matrix[5] is used for the fluid equations. The Roe matrix was developed in the early 80’s and is commonly used in finite volume solvers. The Roe matrix allows the use of a simplified conservative algorithm for the fluid equations.


Integral Equations and Discretization

•The fluid equations and Maxwell’s equations are specific examples of the general conservation law which is written.

•The conservation law is integrated to obtain the integral form

•Using the divergence theorem the volume integral is transformed into an area integral from which the discretization is obtained.

•The implicit discretization is

•The explicit discretization is

( ) ( )QQFtQ Ψ=⋅∇+

∂∂

( ) ( ) dVQdVQFdVtQ ⋅Ψ=⋅⋅∇+⋅

∂∂

∫∫∫

( ) 12/12/1

1 ~~ +−+

+ Ψ⋅Δ=−Δ+− ni

ni

ni

ni

ni tFF

VAtQQ

( ) ( ) dVQadQFdVtQ ⋅Ψ=⋅+⋅

∂∂

∫∫∫

( ) ni

ni

ni

ni

ni tFF

VAtQQ Ψ⋅Δ=−Δ+− −+

+2/12/1

1 ~~


Implicit Source

When the system is stiff an implicit technique is necessary. The implicit solution is updated using a Newton iteration

( ) ( ) ki

ni

ni

ni

ki FF

VAQQ

tQk

iQ

ki

tΨ+−−−

Δ−=Δ⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

Ψ∂+

Δ −+ 2/12/1~~11

•The implicit update is very fast because the Jacobian is a simple algebraic function of the conserved variables. As a result, no numerical Jacobian needs to be calculated. Furthermore the implicit step only requires a small matrix solution for each cell.

•At every step the electromagnetic field and the electron fluid are updated implicitly while the ions are updated every 20 steps or so. The result is that at every step a 7X7 system is solved for each cell to update the electromagnetic field and the electron fluid and every 20 steps an 11X11 system is solved for each cell when the ion fluid is updated as well.

•Faraday’s law and the continuity equations are solved explicitly because they are homogenous equations.

•The solution to the linear system for each cell is performed using about 3 symmetric GaussSeidel iterations for every Jacobian step.


Implicit Setup

The Jacobian for the full 11X11 implicit update follows. Recall that all homogenous equations are updated explicitly. The implicit step is performed for each cell. In stiff problems most of the time is spent doing the implicit update so a quick implicit step is important.

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−−

−−−−−

−−−

−

−−

−

=∂Ψ∂

000000000000000000000000000

0000000000000

0000000000000000

0000000000000

0000000000000000

ie

ie

ie

iziiiyiiixiiziyixi

iixiyi

iixizi

iiyizi

ezeeeyeeexeezeyexe

eexeye

eexeze

eeyeze

Q

rrrr

rrUrUrUrErErErrBrBr

rBrBrrBrBr

UrUrUrErErErrBrBr

rBrBrrBrBr

ρρρρ

ρρ

ρρρρ

ρρ

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

zEyExEiezUi

yUixUieezUe

yUexUe

Qρρρ

ρρρ

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−−−−

⎟⎠

⎞⎜⎝

⎛ ++

⎟⎠

⎞⎜⎝

⎛ −+

⎟⎠⎞⎜

⎝⎛ +−

⎟⎠

⎞⎜⎝

⎛ −+

⎟⎠

⎞⎜⎝

⎛ ++

⎟⎠

⎞⎜⎝

⎛ −+

⎟⎠⎞⎜

⎝⎛ +−

⎟⎠

⎞⎜⎝

⎛ −+

=Ψ

ereiriereiriereiri

zEizUyEiy

UxEixUiri

xBiyUyBix

UxEiri

xBizUzBix

UyEiri

yBizUzBiy

UxEiri

zEezUyEey

UxEexUere

xBeyUyBex

UxEere

xBezUzBex

UyEere

yBezUzBey

UxEere

ρρρρρρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

i

ii mqr =

e

ee mqr =where and


Explicit Source

When the source terms are not stiff the source can be updated explicitly.

•The explicit update is slightly faster than the implicit update when the system is not stiff.

•For non-stiff problems, most of the calculation time is spent calculating the numerical fluxes.

•For stiff problems the explicit solution quickly blows up while the implicit solution remains stable.

•In order to speed up the algorithm we have tried updating the electromagnetic field explicitly for several steps and then doing an implicit update with the fluid equations, however, we have found that for even moderately stiff problems this technique can produces un-physical oscillations and frequently leads to amplifying solutions which eventually blow up.

( ) ni

ni

ni

ni

ni tFF

VAtQQ Ψ⋅Δ=−Δ+− −+

+2/12/1

1 ~~


Roe Matrix

A discrete Jacobian can be calculated for many conservation laws and is often useful in designing efficient conservative algorithms. For the Maxwell system the discrete Jacobian is the same as the exact Jacobian. A Roe matrix is used for the discrete Jacobian of each of the fluid systems. The Roe matrix satisfies the following properties.

[ ] [ ]RLRL QQAFF −=− ~

A~

A conservation law can be written 0=∂∂+

∂∂

xQA

tQ

withxQA

xF

∂∂=

∂∂

whereQFA

∂∂= Is the Jacobian.

FQ→A~ maps

( )QAAQQ RL →→ ~ ,As

A~The eigenvectors of are linearly independent


Numerical Flux

The Numerical Flux is calculated according to LeVeque[3] in the following way.

ki

ki

ki

k

ki

ni rw

xtF 2/12/12/12/12/1

~121~

+++++ ⋅⋅⎥⎦⎤

⎢⎣⎡

ΔΔ−⋅= ∑ λλ

( )θSww ki

ki ⋅= ++ 2/12/1

~

( ) ( )θθ ,1modmin=S

ki

kI

ww

2/1+

=θ 0 if 2/1 2/1 >−= +kiiI λ 0 if 2/3 21 <+= +

k/iλiI

And the minmod flux limiter

Using the limited eigenwave

Where and or

k/iλ 21+ Is the kth eigenvalue of the discrete Jacobian calculated at I+1/2

k/iw 21+ Is the kth eigenwave of the discrete Jacobian calculated at I+1/2

k/ir 21+ Is the kth right eigenvector of the discrete Jacobian calculated at I+1/2


Langmuir WavesAs part of the code validation we have tested the algorithm on a simple electrostatic wave. The oscillation frequency and amplitude agree with that predicted by linear theory. The explicit solver produces slowly amplifying waves while the implicit solver produces a slowly decaying wave.

In a shock problem very similar to the one described on the next page without the B field, a Langmuir wave propagates from the shock a short time into the simulation. Notice the wave steepening that appears in the waves in the right half of the shock.


Brio and Wu MHD Shock[1]

Ideal MHD initial conditions

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

++++++

+

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

z

y

x

ei

ei

xeexii

ei

xeexii

ei

xeexii

ei

z

y

x

z

y

x

B

B

B

pp

UU

UU

UU

BBBpUUU

ρρρρρρρρρρρρ

ρρ

ρ

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

××

×

×

×

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−

−

−

−

−

0101105.7

0001050001105000101

2

3

5

5

3

z

y

x

z

y

x

i

iz

iy

ix

i

e

ez

ey

ex

e

BBBEEEpUUU

pUUU

L

ρ

ρ

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

×−×

×

×

×

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−

−

−

−

−

0101105.7

000105000125.0105000

10125.0

2

3

6

6

3

z

y

x

z

y

x

i

iz

iy

ix

i

e

ez

ey

ex

e

BBBEEEpUUU

pUUU

R

ρ

ρ

The initial conditions are given in terms of primitive two fluid variables. These conditions are equivalent to the MHD initial conditions given on the left.

Ideal two fluid initial conditionsL is the initial condition on the left half of the domain and R is the initial condition on the right half of the domain. The initial conditions are chosen so that all MHD waves travel at roughly the same speed.

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

××

×=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−

−

−

0101105.7

1010001

2

3

4

z

y

x

z

y

x

BBBpUUU

L

ρ

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

×−×

×=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−

−

−

0101105.7

101000

125.

2

3

5

z

y

x

z

y

x

BBBpUUU

R

ρ


MHD Shock Scaling The initial conditions of the previous page suggest that there is one unique two fluid solution to the Brio and Wu MHD shock. Here we emphasize that it is not the initial conditions which are important, but the MHD wave speeds. One can imagine changing the plasma size or increasing the magnetic field and the number density while keeping the MHD wave speeds the same. This amounts to nothing more than changing the Debye length and gyro radius while keeping their ratio the same. The structure of the Brio and Wu MHD shock is completely determined by the Debye length and gyro radius provided the species mass ratio is the same and the species charge to mass ratio is the same. In the following, the non-dimensionalization for the two fluid shock with variable Debye length and gyro radius is presented. The Debye length is d and the ion gyro radius is r.

( )BαUErrαPαUαUρt

αUαρ ×+⎟⎟⎠

⎞⎜⎜⎝

⎛=∇+⋅∇+∂

∂ααρ1

( ) αραααα

αα UErr

PeUte

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=+⋅∇+∂

∂ 1

• Species Momentum

• Species Energy

• Ampere’s Law

• Poisson’s Equation

( )iiieee UrUrd

BtE

ρρ +⎟⎟⎠

⎞⎜⎜⎝

⎛ ×−=×∇−∂∂ −2101

( )iiee rrd

E ρρ +⎟⎟⎠

⎞⎜⎜⎝

⎛ ×=⋅∇−2101

The electron charge to mass ratio is and ion charge to mass ratio is and ion to electron mass ratio is 1000. It should be clear now that if the Debye length and gyro radius are smallthe problem becomes stiffer and we say the fluid-electromagnetic coupling is strong. When the the Debye length and gyro radius are large the problem becomes less stiff and the fluid-electromagneticcoupling is weak.

3101 −×−=er 1=ir


Two Fluid Shock, MHD limit In the density plot the bump on the right is the alfven wave, and the rarefaction wave on the left is the fast magnetosonic wave. Both plots match the MHD solution well. Reducing the grid resolution to 1000 still gives decent results as can be seen in the shock comparison later on. This seems to suggest that for these problems it is not necessary to resolve the Debye length.

In the following we have used a small Debye length and gyro radius to try and match the MHD solution to the Brio and Wu shock. As can be seen below much of the shock structure is captured despite having finite Debye length and gyro radius. The time step for this simulation is roughly 0.5 times the plasma frequency. For stiff problems the time resolution is dominated by the plasma frequency not the speed of light. Notice that each grid cell is roughly 8 Debye lengths and 0.08 ion gyro radii. There are 4000 grid cells.


Two Fluid Shock, MHD limit

The MHD equations do not predict a bulk velocity in the Z direction. In the two fluid case, the electrons have inertia and the electron current supports most of the magnetic field. Consequently a bulk Z velocity is predicted by the two fluid equations.

The MHD equations do not predict a Z magnetic field. However, in the two fluid system, the Bz is induced by a Jy which is caused by the Jz crossed with the Bx. Including the hall term and the rate of change of current allows the z magnetic field to be resolved in the two fluid system.


Two Fluid Shock, MHD limit – varying grid

Here we have changed the grid resolution on the same problem as on the previous page, d=3.16e-5, g=3.16e-3. At a grid resolution of 500 most of the structure that appears at a grid resolution of 4000 is resolved. From Debye length alone, one might expect that the solution would not be well resolved until the domain is split into about 30,000 grid cells, but it appears that we can avoid resolving the Debye length scales and still gain most of the MHD structure with added two fluid physics.


Two Fluid Shock, MHD limit – varying gridCurrent spikes which support the magnetic shocks are much better resolved at higher resolutions. The lower resolutions do a good job of matching the large wavelength structure. In the two fluid system, for very stiff problems, not only do we need to be able to resolve shock discontinuities, we also need to be able to resolve current sheets which travel perpendicular to the shock. At high resolution, a current sheet is clearly resolved at 0.6 x units in the plot of vz.


Two Fluid Shock, MHD limit – varying couplingThe two fluid system admits a continuous range of solutions for the Brio and Wu MHD shock because the coupling can be adjusted while keeping the MHD wave speeds the same. This plot shows a sequence of two fluid solutions to the MHD shock problem. It is apparent that increasing the coupling improves the agreement between the MHD solution and the two fluid solution.

As stated previously the vz differs substantially because in the ideal MHD solution the current is represented by the curl of the magnetic field and in the two fluid system the current is the combination of the electron and ion currents. The two fluid system predicts strong dependence of vz on the coupling. 1000 grid cells were used in these comparison simulations.


Gas Dynamic Shock The Debye length and gyro radius are increased to the point that the electromagnetic field is decoupled from the fluid and the two fluids move independently. The electron shock propagates at the electron acoustic speed and the ion shock at the ion acoustic speed, the plot below plots the electron fluid density at 1/40 the time that the ion density is plotted. Both species travel slower than the speed of light with the electrons traveling at a maximum of 0.3c.

The shock in B produces a shock in E and the electromagnetic wave quickly propagates out of the domain at the speed of light without any interaction with the fluid.


Two Fluid Shock, Gas Dynamic limitIn these simulations we decrease the coupling in the Brio and Wu shock problem to the point that we get gas dynamic shocks. The MHD density profile which is calculated as the weighted average of the electron density and the ion density is plotted below. The density profile varies wildly as Debye length and gyro radius are changed.

As the coupling is increased, less and less of the B shock is supported by the displacement current and more is supported by the fluid current. In particular only high frequency waves are supported by the displacement current. This is the assumption that is made in MHD, and the validity of this assumption is demonstrated in the two fluid solution below and on the next page.


Two Fluid Shock, Gas Dynamic limitAs the coupling is increased higher and higher frequencies of the B shock are frozen into the fluid. In the plot below it is apparent that for weak coupling, all the electromagnetic waves travel near the speed of light. As the coupling is increased the low frequency waves travel slower than the high frequency waves until the point that the B shock is “frozen” into the fluid.

Notice that on the high density side of the shock on both plots the coupling is stronger and electromagnetic waves are higher frequency than on the low density side. In both cases the high frequency electromagnetic waves travel faster than the low frequency waves. The plot below also demonstrates that as the coupling is increased the importance of the displacement current is reduced.


Extension to 3 Dimensions and Further Work

1. The stiff problems are time consuming and this will become more important in 3 dimensions. Resolving the speed of light makes the system computational intensive, but resolving the plasma frequency can make the problem even harder to solve.

2. Explicitly satisfying Poisson’s equation will be important; we are currently working on methods to deal with this constraint.

3. The magnetic flux constraint will need some consideration in 3 dimensions.

Further work

1. 3 dimensional approximate Riemann solvers for fluid equations are well understood and are commonly used in industry and academia. The extension to 3 dimensions is simple because the same solver can be used except now a one dimensional problem must be solved across each cell face.

2. Approximate Riemann Solvers for Maxwell’s equations with source terms are relatively new[4]. Another graduate student, Chris Aberle, has shown that simple first order approximate Riemann solvers for Maxwell’s equations with source terms work on triangular meshes. It is believed that higher order methods will work on rectangular meshes.

3. In 3 dimensions the source terms can still be updated implicitly, cell by cell, one implicit update per cell. As a result the implicit computation time will not increase when the system is extended to 3 dimensions.

Extension to 3 Dimensions


Summary & Conclusions

Implicit and Explicit schemes of a one-dimensional approximate Riemann solver for the non-relativistic ideal two fluid plasma system have been developed.Traditional techniques of fluid dynamics including the Roe matrix and flux limiters have been applied to the two fluid system.The two fluid equivalent of the Brio and Wu MHD shock has been described and tested allowing for varying fluid-electromagnetic coupling.The same algorithm has been shown to produce correct solutions to electrostatic, electrodynamic, gas dynamic and MHD-like problems.We have discussed how this solver can be extended to multiple dimensions.


References

[1] M. Brio and C.C. Wu, An Upwind Differencing Scheme for the Equations of Ideal Magnetohydrodynamics, JCP 75 1998, pg 400-422

[2] Ami Harten, High Resolution Schemes for Hyperbolic Conservation Laws,JCP 49, 1983 pg 357-393

[3] Randall J LeVeque, Finite Difference Methods for Differential Equations, Lecture Notesfor AMATH 585-586, University of Washington

[4] C.D. Munz, R. Schneider, and U. Vos, A Finite-Volume Method for the Maxwell Equations in the Time Domain, Siam Journal on Scientific Computing 22 (2000), no. 2, 449-475.

[5] P.L. Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference SchemesJCP 135 (reprinted 1997) pg 250-258

a finite volume scheme for the two fluid plasma system

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