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

Mass and energy conservative high order diagonally implicit Runge--Kutta schemes for nonlinear Schr\"odinger equation in one and two dimensions