Imagine there is a 2dim area and you bring some fixed charges (of
finite size) in there. As a consequence, a non-constant
electrostatic potential will establish.
In the empty parts, this potential follows the Laplace equation
with the boundary conditions imposed by the fixed charges.
The Laplace equation (Laplace phi(x,y)=0 whith
phi(x,y) being the potential at position (x,y)) is a
partial differential equation (PDE).
Numerical solving of PDEs is not as simple as solving ODEs. In order to
solve the Laplace equation, this program divides up the area into little
elements (discretisation of the problem; finite element method).
The second partial derivation in the Laplace equation is computed
numerically using standard difference quotient of a cell with its
4 neighbour elements. This results in a quite large but sparse linear
system (number of rows and columns of the Matrix is the number of floating
(non-fixed) elements in the area and hence at least several thousand).
In order to solve this system, iterative the following methods are
Gauss-Seidel algorithm: This is the standard iterative algorithm
for solving sparse diagonally-dominant linear systems.
Thomas-TDMA (tridiagonal matrix algorithm): In contrast to
Gauss-Seidel, this algorithm computes whole lines of successive
floating elements exactly and iterates over these line chunks.
Successive overrelaxation is implemented for both these algorithms
which can greatly improve the speed of convergence when a suitable
factor is entered manually (see "omega" setting).
The Thomas-TDMA will normally converge faster than the Gauss-Seidel
algorithm but it does not allow for periodic boundary conditions
(since this would destroy the tridiagonal structure of the system).
Okay, so if you think all that GUI-stuff is self-explaining and just want
to get on with it, just note the following: Keys '+' and '-' can
be used to zoom in and out and dragging with the right mouse
button pressed will pan the simulation area.
Description of the GUI elements:
Solver: allows you to select the solver to use; see above.
Timeslice: Time in msec the solver may run until processing user
events (like adding charges with the mouse) are checked again.
Note that the solver can only be interrupted after complete iterations
and for large area sizes, these may take longer than the specified
Epsilon: The two spin boxes allow you to set the PDE solver
convergence threshold epsilon. In the screen shot above,
"1" and "-6" means epsilon=1e-6.
The PDE solver will stop iterations when the largest change of a
potential value during the current iteration is smaller than
Omega: This is the overrelaxation factor which is in range
1..2 (hence 100%..200%). A value of 100% (displayed as "off") switches
off this feature and results in normal solver iterations.
Values larger than 1 result in successive overrelaxation which will,
when a proper value is set, speed up solver convergence considerably.
(Watch the dots in the terminal for that.)
Shuffle: Normal Gauss-Seidel or TDMA will iterate through the
linear equation system always in the same way (i.e. top-down).
When setting "shuffle" to something else than "[none]", the solvers
will run lines top-down and bottom up and use linear systems
generated via X and Y sweeps.
Note: Experiments showed that this will often slow down convergence a bit.
MaxIter: Limit iterations of the solver. Forget it and leave it
at "[no limit]". This is only useful for testing.
Periodic BC: Whether to use periodiv boundary conditions (i.e.
torus-like area) or not.
Note that the TDMA solver cannot handle periodic BC and hence this
check box is enabled only for the Gauss-Seidel solver.
Start/Stop: Start and stop the PDE solver. When stopped, the
button displays "Start" and the PDE solver will not do anything.
When running (the button displays "Stop"), the PDE solver will start
running whenever something changes and will stop temporarily when
the algorithm has converged. It will, however, re-start automatically
once one changes some charge etc.
Area... openes the area size dialog to change the number of
elements in the area.
Reset will fill the complete area with fixed or floating
(depending what you specified as "fill type" in the area dialog; default
is floating) elements of potential 0.
Zero will set the potential value of all floating (non-fixed)
elements to 0.
NColors allows you to change the number of colors in the display.
The color display will update automatically so that the lowest and
highest potential value fit into color range. It is, however, sometimes
interesting to limit the number of colors to "see more". See below
for an example image.
Shape: Selects the shape you can draw with the left mouse button.
Type: Selects the type of mouzse-drawn elements.
Potential: Choose electrostatic potential for elements drawn with