A new Lagrange multiplier approach for gradient flows: A new Lagrange multiplier approach

We propose a new Lagrange multiplier approach to design unconditional energy stable schemes for gradient flows. The new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed SAV approach (Shen, Xu, and Yang 2018), but enjoys two additional advantages: (i) schemes based on the new approach dissipate the original energy, as opposed to a modified energy in the recently proposed SAV approach (Shen, Xu, and Yang 2018);and (ii) they do not require the nonlinear part of the free energy to be bounded from below as is required in the SAV approach. The price we pay for these advantages is that a nonlinear algebraic equation has to be solved to determine the Lagrange multiplier. We present ample numerical results to validate the new approach, and, as a particular example of challenging applications, we consider a block copolymer (BCP)/coupled Cahn–Hilliard model, and carry out new and nontrivial simulations which are consistent with experiment results.