Efficient modified techniques of invariant energy quadratization approach for gradient flows

Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.