2d cold-ion equations

2D Cold-Ion EquationsDiscontinuities in its Solutions

James Cheung

Department of Scientific ComputingFlorida State University

Tallahassee, Florida 32306

December 4, 2013

Overview

I The Cold-Ion Equations

I Numerical Methods

I Verification

I Some cool pictures

The Cold-Ion Equations

∂n

∂t+∇ · (n~v) = 0

∂~v

∂t+ ~v · ∇~v = −∇φ

∇2φ = eφ − n

I n denotes the plasma density

I ~v denotes the flow velocity

I φ denotes the electrostatic potential

An Explicit Form

∂n

∂t+

∂

∂x(nu) +

∂

∂y(nv) = 0

∂u

∂t+

∂

∂x(

u2

2+ φ) + v

∂u

∂y= 0

∂v

∂t+

∂

∂y(

v 2

2+ φ) + u

∂v

∂x= 0

∂2φ

∂x2+∂2φ

∂y 2= eφ − n

What are plasmas?

I A plasma is a gas that has enough energy to seperateelectrons from its respective atoms.

I A plasma is a mixture of electron gas and positive ions.

I Plasmas are often considered collisionless, which led to thedevelopment of the Vlasov-Maxwell and Vlasov-Poissonequations.

I Plasmas are often super-critical fluids.

I Cold plasmas are not super-critical, and the electrontemperature is negligible.

I Cold plasmas can be modelled using the Cold-Ion Equations.

What do these equations model?

I These equations are representative of the cold-ion limit ofcollisionless plasma modelled by the Vlasov-Poisson equations.

I Instead of determining a distrubution of ion-densities at everypossible velocity and space coordinate, there is only 1 uniquedensity associated to 1 velocity and 1 spatial coordinate.

I Instead of using Maxwell’s equations to model the behavior ofthe charged nature, the Poisson equation is used instead asthe zero magnetic field limit. This assumption is valid becausein a cold-ion the charge separation between the electron andits ion is very small.

I This model can be viewed as the Euler equations coupled withelectrostatic phenomena.

Why are these equations interesting? (i.e. Focus of myresearch)

With initial conditions in density that contain step-discontinuities,an infinte spike forms at approximately x ≈ t. This has beenshown both numerically and analytically using the stationary phasemethod for the one dimensional case. (See Perego, et. al.)

Overview


I Numerical Methods

I Verification


Numerical Methods

I The elliptic PDE was solved using finite differences withNewton’s method to deal with the nonlinearity

I The hyperbolic system was solved using an explicit staggeredLax-Friedrichs finite difference scheme.

Staggered Lax-Friedrichs

The scheme is a two-step scheme. For the transport equation, it isgiven as:

Un+ 1

2

i+ 12,j+ 1

2

=1

4

(Un

i ,j + Uni+1,j + Un

i ,j+1 + Uni+1,j+1

)− ∆t

2∆x

(Un

i+1,j − Uni ,j

)− ∆t

2∆y

(Un

i ,j+1 − Uni ,j

)

Un+1i ,j =

1

4

(U

n+ 12

i− 12,j− 1

2

+ Un+ 1

2

i− 12,j+ 1

2

+ Un+ 1

2

i+ 12,j− 1

2

+ Un+ 1

2

i+ 12,j+ 1

2

)− ∆t

4∆x

(U

n+ 12

i+ 12,j− 1

2

− Un+ 1

2

i− 12,j− 1

2

+ Un+ 1

2

i+ 12,j+ 1

2

− Un+ 1

2

i− 12,j+ 1

2

)− ∆t

4∆x

(U

n+ 12

i− 12,j+ 1

2

− Un+ 1

2

i− 12,j− 1

2

+ Un+ 1

2

i+ 12,j+ 1

2

− Un+ 1

2

i+ 12,j− 1

2

)

Numerical Error

The theoretical truncation errors:

I Second Centered Difference: O(∆x2 + ∆y 2

)I Two-Step Lax-Friedrichs: O

(∆x2+∆y2

∆t + ∆x + ∆y + ∆t)

In addition to truncation error, we have phase errors for theTwo-Step Lax-Friedrichs scheme. The scheme is dissipative anddispersive.

Why do I want to use such a “terrible” method?

I I am not particularly interested in achieving an extremelyprecise solution

I The numerical dissipation allows the algorithm not to blow upwhen the infinite spike forms.

I The dipersion error is smoothed out by using a staggered grid.

I The scheme perserves monotonicity, whereas higher orderschemes, i.e. Lax-Wendroff does not. (Godunov’s Theorem)

I The phase errors can be reduced by taking timesteps close tothe theoretical limit of stability.

Overview


I Numerical Methods

I Verification


Verification

Because the Poisson equation convergence rates were studied todeath in the lab, I will focus my presentation on verifying that mycode is consistent with the properties of the staggeredLax-Friedrichs method. The test problem is the 2D transportequation.

∂u

∂t+∂u

∂x+∂u

∂y= 0

u (x , y , 0) =

{1 if |x | < 5 and |y | < 5

0 otherwise

Verification (continued)

The numerical rates of convergence

Numerical Error and Rates of Convergence

h ||u(xi , yj , tn)− Un

i ,j ||2 rate

1/4 0.29521/8 0.2321 0.3469

1/16 0.1856 0.32251/32 0.1501 0.30631/64 0.1224 0.2943

The numerical rate of convergence is low due to the phase errorcaused by the numerical method. ∆t was taken to be h

2

Visual Verification

These pictures represent the numerical solution at t = 5.

Figure: Solutions of the test equation. ∆t = 0.1h, ∆t = 0.5h, and ExactSolution

The numerical solutions verify that my algorithm is dissipative andthat it does not create spurious oscillations.

Overview


I Numerical Methods

I Verification


Some Cool Pictures

Three initial conditions for the PDE will be considered. Thesecases model the expansion of cold-ion plasma of a higher densityinto a cold-ion plasma of a lower density.

I The 2D analogue of the initial conditions studied in Perego,et. al.

I A square of high density

I A circle of high density

It will be assumed that an infinite domain is adequatelyrepresented using homogeneous Neumann boundary conditions.

2D Analogue

n (x , y , 0) =

{1 if x ∈ (−∞, 0) and y ∈ (−∞,∞)

0.25 otherwise

~v (x , y , 0) = 〈0, 0〉

A Square of High Density

n (x , y , 0) =

{1 if ||x || < 5 and ||y || < 5

0.25 otherwise

~v (x , y , 0) = 〈0, 0〉

A Circle of High Density

n (x , y , 0) =

{1 if x2 + y 2 < 25

0.25 otherwise

~v (x , y , 0) = 〈0, 0〉

Algorithm

While final time has not been reached:

1. Solve for φ on base grid.

2. Solve for n and ~v on staggered grid.

3. Solve for φ on staggered grid.

4. Solve for n and ~v on base grid.

5. Timestep

This algorithm is expensive. Newton’s method must be used tosolve the nonlinear Poisson equation for φ twice per time step.Remember, ∆t ≤ h

2 . Thus, in these simulations a gridsize h = 0.1and ∆t = 0.4 was used. In my 1D simulations, it was found that agridsize of h = 0.02 was ideal for modelling the phenomena. Thiswould be cost-prohibitive for my current 2D algorithm.

2d cold-ion equations

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