  The real-time contour formalism for Green's functions provides time-dependent
information of quantum many-body systems. In practice, the long-time simulation
of systems with a wide range of energy scales is challenging due to both the
storage requirements of the discretized Green's function and the computational
cost of solving the Dyson equation. In this manuscript, we apply a real-time
discretization based on a piece-wise high-order orthogonal-polynomial expansion
to address these issues. We present a superconvergent algorithm for solving the
real-time equilibrium Dyson equation using the Legendre spectral method and the
recursive algorithm for Legendre convolution. We show that the compact high
order discretization in combination with our Dyson solver enables long-time
simulations using far fewer discretization points than needed in conventional
multistep methods. As a proof of concept, we compute the molecular spectral
functions of H$_2$, LiH, He$_2$ and C$_6$H$_4$O$_2$ using self-consistent
second-order perturbation theory and compare the results with standard quantum
chemistry methods as well as the auxiliary second-order Green's function
perturbation theory method.
