  In the present paper, integrable semi-discrete and fully discrete analogues
of a coupled short pulse (CSP) equation are constructed. The key of the
construction is the bilinear forms and determinant structure of solutions of
the CSP equation. We also construct Nsoliton solutions for the semi-discrete
and fully discrete analogues of the CSP equations in the form of Casorati
determinant. In the continuous limit, we show that the fully discrete CSP
equation converges to the semi-discrete CSP equation, then further to the
continuous CSP equation. Moreover, the integrable semi-discretization of the
CSP equation is used as a selfadaptive moving mesh method for numerical
simulations. The numerical results agree with the analytical results very well.
