  The discretization approximation method commonly used to simulate the open
dynamics of system coupled to the environment in continuum often suffers from
the recurrence. To address this issue, this paper proposes a noval
generalization of the discretization approximation method in the complex plane
using complex Gauss quadratures. The effective Hamiltonian can be constructed
by this way, which is non-Hermitian and demonstrates the complex energy modes
with negative imaginary part, describing accurately the dissipative dynamics of
the system. This method is applied to examine the dynamics in two exactly
solvable models: the dephasing model and the single-excitation open dynamics in
the Aubry-Andr\'{e}-Harper model. This approach not only significantly reduces
recurrence and improve the effectiveness of calculation, but also provide the
microscopic viewpoint on the dynamics of system through the effective
Hamiltonian. In addition, a simple relationship between the parameters in
computation and the effectiveness of evaluation is also established.
