A new Lagrange Multiplier approach for gradient flows

Abstract

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 \cite{SAV01}, 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 \cite{SAV01}; 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 applications, we consider a coupled Cahn-Hilliard model for block copolymers (BCP), and carry out interesting simulations which are consistent with experiment results.

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• Liquid-vapour phase behaviour of a symmetrical binary fluid mixture
• A new class of efficient and robust energy stable schemes for gradient flows
• The Phase Field Method for Geometric Moving Interfaces and Their Numerical Approximations

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• A novel Lagrange Multiplier approach with relaxation for gradient flows
• Supplementary Variable Method for Developing Structure-Preserving Numerical Approximations to Thermodynamically Consistent Partial Differential Equations
• Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn-Hilliard systems with bounded mass source
• Strang splitting structure-preserving high-order compact difference schemes for nonlinear convection diffusion equations
• Global constraints preserving SAV schemes for gradient flows
• A hybrid phase field method for fluid-structure interactions in viscous fluids
• Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise
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• A Relaxed Lagrange Multiplier Approach for Phase Field Models
• Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation
• Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems
• Thermodynamically Consistent Algorithms for Models of Diblock Copolymer Solutions Interacting with Electric and Magnetic Fields