Fundamentals of Plasma Physics

26 Chapter 1. Basic concepts

which may be solved to give

ynew=yold+ ∆tf(yold,t)

Simple and appealing as this is, it does not work since it is numerically unstable.

However, if we use the following scheme we will get a stable result:

ynew−yold

∆t

=f((ynew+yold)/ 2 ,t) (1.46)

In other words, we have used the average of the new and the old values ofyin the

RHS. This makes sense because the RHS is a function evaluated at timetwhereas

ynew=y(t+∆t/2)andyold=y(t−∆t/2). If Taylor expand these last quantities are

Taylor expanded, it is seen that to lowest ordery(t) = [y(t+∆t/2)+y(t−∆t/2)]/ 2.

Sinceynewoccurs on both sides of the equation we will have to solve some sort of

equation, or invert some sort of matrix to getynew.

Start with

m

dv

dt

=q(E+v×B).

Define, the angular cyclotron frequency vectorΩ=qB/mand the normalized electric

fieldΣ=qE/mso that the above equation becomes

dv

dt

=Σ+v×Ω (1.47)

Using the implicit scheme of Eq.(1.46), show that Eq. (1.47) becomes

vnew+A×vnew=C

whereA=Ω∆t/ 2 andC=vold+∆t(Σ+vold×Ω/2). By first dotting the above

equation withAand then crossing it withAshow that the new value of velocity is

given by

vnew=

C+AA·C−A×C

1+A^2

.

The new position is simply given by

xnew=xold+vnewdt

The above two equations can be used to solve charged particle motion in complicated,

3D, time dependent fields. Use this particle integrator to calculatethe trajectory of

an electron moving in crossed electric and magnetic fields where the non-vanishing

components areEx= 1volt/meter andBz= 1Tesla. Plot your result graphically on

your computer monitor. Try varying the field strengths, polarities, and alsotry ions

instead of electrons.

8. Use the leap-frog numerical integration scheme to demonstrate the Rutherford scat-
   tering problem:
   (i) Define a characteristic length for this problem to be the impact parameter for a 90
   degree scattering angle,bπ/ 2. A reasonable choice for the characteristic velocity is
   v∞. What is the characteristic time?