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Recently, a new Lagrange multiplier approach was introduced by Cheng, Liu andShen in \cite{cheng2020new}, which has been broadly used to solve variouschallenging phase field problems. To design original energy stable schemes,they have to solve a nonlinear algebraic equation to determine the introducedLagrange multiplier, which can be computationally expensive, especially forlarge-scale and long-time simulations involving complex nonlinear terms. Thispaper presents an essential improved technique to modify this issue, which canbe seen as a semi-implicit combined Lagrange multiplier approach. In general,the new constructed schemes keep all the advantages of the Lagrange multipliermethod and significantly reduce the computation costs. Besides, the newproposed BDF2 scheme dissipates the original energy, as opposed to a modifiedenergy for the classical Lagrange multiplier approach in \cite{cheng2020new}.We further construct high-order BDF$k$ schemes based on the new proposedapproach. In addition, we establish a general framework for extending ourconstructed method to dissipative systems. Finally several examples have beenpresented to demonstrate the effectiveness of the proposed approach.

An enhanced and highly efficient semi-implicit combined Lagrange multiplier approach with preserving original energy law for dissipative systems