A new Lagrange multiplier approach for constructing structure preserving schemes, III. Bound preserving and energy dissipating

In the third part of this series, we continue to explore the idea of the Lagrange multiplier introduced in the first part [2020, Comput. Methods Appl. Mech. Engr., 391, 114585] and refined in the second part [2022, SIAM J. Numer. Anal., 60, 970-998] to further develop efficient and accurate numerical schemes that preserve the maximum bound principle (MBP) and energy dissipation for solving gradient flows. The proposed framework allows us to begin with any conventional scheme as a predictor step which is followed by two consecutive correction steps written in the form of the Karush-Kuhn-Tucker conditions for structure preserving. The preservation of both energy dissipation and MBP and the solvability of the general resulting scheme are rigorously established. In such a framework, we implement an explicit and efficient scheme by employing the Runge-Kutta exponential time differencing scheme as the predictor step, and give its convergence analysis. Extensive numerical experiments are provided to validate the effectiveness of our approach.