Air Shower Simulations in a Hybrid Approach using Cascade Equations
  A new hybrid approach to air shower simulations is described. At highest energies, each particle is followed individually using the traditional Monte Carlo method; this initializes a system of cascade equations which are applicable for energies such that the shower is one-dimensional. The cascade equations are solved numerically down to energies at which lateral spreading becomes significant, then their output serves as a source function for a 3-dimensional Monte Carlo simulation of the final stage of the shower. This simulation procedure reproduces the natural fluctuations in the initial stages of the shower, gives accurate lateral distribution functions, and provides detailed information about all low energy particles on an event-by-event basis. It is quite efficient in computation time. 
Hans-Joachim Drescher and Glennys R. Farrar