R&D of Mesoscale Particle Simulation Code

** For high-temperature plasma simulations, one needs
several types of codes to handle macroscopic (large scale)
and microscopic (small scale) phenomena. The mesoscale
particle code is aimed to deal with plasma kinetic
phenomenon both with medium time and space scales.
Why Mesoscale Particle Code ?
Since we have to keep track of particles motion along their orbits,
we start with a particle code. However, a plasma has an eigenmode
called "plasma" (Langmuir) oscillations, whose time and space scales
are of the plasma frequency and the Debye length, respectively, which
need be resolved (even if one is not interested in them) for stable
numerical simulations. Great deal of efforts of removing these small-
scale oscillations have been taken in the United States in early 1980s,
especially in Los Alamos and Livermore National Laboratories.
The former group makes use of fluid equations and the backward
time discretization technique to filter out plasma oscillations, where
particle contributions are counted as the first and second moment
source terms to the equations. The latter group devised high accuracy
**numerical lowpass filter to plug in to a particle code. The pros and

cons of these methods are numerical stability and substantial

computation load for the former, and simplicity of the algorithm but

numerical instability for the latter.

What is New with My Algorithm ?

Since I started developing a mesoscale particle code, I could adopt

numerically stable backward time discretization technique. My

invention is to represent all unknown (future) field quantities in the

equations by the already known quantities with aid of the Maxwell and

Newton equations of motion. This rendered use of minimum a priori

assumptions in equations, numerical stability in general, and less

computational load than with the Los Alamos code.

The derived field equations to be solved are fully implicit ones in

which unknown quantities appear on the leftside of the equations. The

size of the unknowns are three times the number of space grids. For

spatial dimensions less or equal to two, the solution converges fairly

well.

However, for three dimensions in space, the number of the

unknowns amounts to 300,000 even if there are only 50 grids along one

axis of the space, which posed numerical difficulty in solving such giant

matrix equations. It was only after I had adopted the bi-conjugate

gradient method which is aimed to solve giant nonsymmetric matrix

equations, that the 3D mesoscale particle code became available for

realistic plasma simulations.

The algorithm and accuracy proof/demonstrations of the mesoscale

particle code that I developed from nearly a scratch (called HIDEN)

are shown in the following papers. The second paper describes three

dimensional cases with numerical techniques required in such

dimensions.

References:

Physics of High-Temperature Plasmas (in Japanese)

(Parity Physics Course, Maruzen; 1991, 1997)

Algorithm and proof of mesoscale particle code

J.Comput.Phys., 79, 209 (1988).

３D Mesoscale particle code

J.Comput.Phys., 107, 124 (1993).