  In this paper, we propose a discrete Hamilton--Jacobi theory for (discrete)
Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we
first provide a novel geometric Hamilton--Jacobi theory for continuous contact
Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian
formulation, we obtain the discrete equations for Hamiltonian dynamics by the
discrete Legendre transformation. Based on the discrete contact Hamilton
equation, we construct a discrete Hamilton--Jacobi equation for contact
Hamiltonian dynamics. We show how the discrete Hamilton--Jacobi equation is
related to the continuous Hamilton--Jacobi theory presented in this work. Then,
we propose geometric foundations of the discrete Hamilton--Jacobi equations on
contact manifolds in terms of discrete contact flows. At the end of the paper
we provide a numerical example to test the theory.
