Time discrete scheme matched adaptive time step for particle transport equations

The conventional time discrete schemes seldom consider adaptive time step. Therefore some physical quantity for time variable exits numerical oscillation. The motivation in this paper is to provide a time discrete scheme which matches adaptive time step. The typical exponential method, diamond difference and modified time discrete scheme, second-order time evolution scheme are researched for adaptive time step. Some numerical results show that time differential curves (particle current especially) are very smooth for second-order time evolution scheme associated the exponential method.