Because every particle in the trap interacts with every other particle, ion trapping simulation falls in the category of long-range interaction problems. The long-range interaction problem is notorious for its computational intensity: it is generally on a sequential machine . If a ``tree-based force calculation" is implemented, computational time can be reduced to on a sequential machine; an adjustable error term controls the tradeoff between speed and accuracy . Utilizing parallel machines should allow for significant improvements in speed without compromising accuracy.
Another computationally intensive algorithm which should show significant speed-up on a parallel machine is that of the Fast Fourier Transform. The Fourier transform, which projects a function onto a set of periodic functions, is useful for indicating whether a function is periodic and for analyzing the frequency components of a periodic signal. The Fourier series, , corresponding to a discrete set of data points, , is defined as :
The Fast Fourier Transform is a algorithm for computing the Fourier series of a set of data points. One goal of the proposed project is to implement the Fast Fourier transform in HPF on a parallel machine.
At least two finite-difference methods will be explored as a means of modeling the time evolution of the ions' positions and velocities. The first method is the Verlet algorithm, which, in its simplest form, uses the particles' positions at the two previous time steps and the forces on the particles to generate a new set of position vectors :
where represents a particle's position vector at time and represents a the force acting on a particle at time . The second method is the Gear predictor-corrector algorithm, which uses a Taylor series approximation to `predict' the position vector at time , then re-evaluates higher-order derivatives using the newly generated position vectors, and finally `corrects' the position vectors accordingly .