Efficient, high-order, and dynamically positive Lagrange multiplier approaches for general dissipative systems

Recently, a new Lagrange multiplier (LM) approach was introduced by Cheng et al. [“A new Lagrange Multiplier approach for gradient flows,” Comput. Method Appl. Mech. Eng. 367, 113070 (2021)] for simulating various gradient flows and has been proved to be very efficient in constructing the scheme with original energy stability. In this paper, we first consider a series of new LM schemes based on Crank–Nicolson and BDFk formulas for general dissipative systems, which not only unconditionally preserve the original dissipation law but also save nearly half computational costs of the original LM scheme. We propose two modified techniques to enhance stability and obtain an accurate numerical solution of ξ with a relatively large time step by adding a stabilized term for the dissipative systems and introducing an artificial parameter ν(t) to the energy evolution equation. Second, to construct linear and energy stable schemes, a dynamically positive Lagrange multiplier scheme is considered to preserve a modified energy stability. We introduce a constant-free auxiliary variable E(t)=ξ(t)E(ϕ) and replace the constant C in the scalar auxiliary variable-based schemes with a flexible function δ(t) to avoid the calculation risk with an unsuitable constant C. We provide rigorous proof of the unconditional energy stability for all constructed schemes. Finally, several numerical experiments are performed to validate the theoretical analysis and the efficiency of the proposed schemes.