Algebraic modeling and parallel computing

In this paper, we present an algebraic approach for beam dynamics simulation in linear and circular accelerators. A two- and three-dimensional distribution function approach is employed within the matrix formalism for Lie algebraic methods and computer algebra codes. Implementing software design provides the code rather good maintainability, reusability, and extensibility. This approach is an alternative to well-known Particle-in-Cell approach. But it can be easily applied to the PIC-approach and thus to make better their effectiveness. As a required demand, the code also include symplectic integration methods (based on a correcting procedure for aberration matrices) up to an approximation order.