Speeding-Up Collision Probability Calculations

Abstract

The requirements for extensive computing time and large computer memory have been recognized as major deficiencies of the collision probability method. This paper presents two methods for speeding up the calculations in the above two areas. One method is concerned with the calculation of collision probabilities, while the other allows a faster solution of resulting systems of linear algebraic equations. In two-dimensional geometry, the collision probabilitiesare expressed as linear combinations of Bickley functions, , the evaluation of which is the main time consumer for small and medium size problems. The new method presented here applies a numerical integration of the polar angle instead of Bickley function calculation. As a result, the algorithm is more robust and twice as fast. The solution of the collision probability equation is usually obtained by direct methods of matrix decomposition. The computing time is proportional to the third degree of the number of unknowns and increases rapidly with the increase of the problem size. To speed-up the calculation, the within-group matrix is subdivided into a number of blocks. The solution is obtained iteratively by block-matrix iteration using the traditional Gauss-Seidel method. Test results show a decrease in computing time by more than one order of magnitude.