  We present and analyze a series of conservative diagonally implicit
Runge--Kutta schemes for the nonlinear Schr\"odiner equation. With the
application of the newly developed invariant energy quadratization approach,
these schemes possess not only high accuracy , high order convergence (up to
fifth order) and efficiency due to diagonally implicity but also mass and
energy conservative properties. Both theoretical analysis and numerical
experiments of one- and two-dimensional dynamics are carried out to verify the
invariant conservative properties, convergence orders and longtime simulation
stability.
