This work is about the effective characterization for stochastic dynamical
systems with tempered stable L\'evy process. To quantify macroscopic or
effective dynamical behaviors of these stochastic systems, we examine two
deterministic tools: mean exit time and probability density evolution, which
are solutions of nonlocal partial differential equations (nonlocal elliptic
equation and nonlocal Fokker-Planck equation) respectively. We develop accurate
numerical methods, together with stability and convergence analysis, to compute
the mean exit time and Fokker-Planck equations associated with these one and
two dimensional stochastic systems. We further illustrate these methods with
numerical experiments in several examples.