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A new Lagrange Multiplier approach for gradient flows
Qing Cheng, Chun Liu, Jie Shen
TL;DR
Gradient-flow schemes need unconditional energy stability without sacrificing accuracy and efficiency. The paper introduces a Lagrange Multiplier approach that dissipates the original energy and removes the lower-boundedness requirement, at the cost of a nonlinear algebraic solve. Numerical results validate its stability, accuracy, efficiency, and application to block-copolymer simulations consistent with experiments.
Problem
Existing SAV schemes dissipate a modified energy and require the nonlinear part of the free energy to be bounded from below.
Method
The approach uses a scalar Lagrange Multiplier to construct unconditionally energy-stable gradient-flow schemes, with efficient linear solves plus a nonlinear algebraic equation.
Results
dE(φ)/dt = −(µ, Gµ), establishing dissipation of the original energy; numerical tests also validate stability, accuracy, and efficiency.
Takeaways & Limitations
The method applies directly to free energies without a lower-bounded nonlinear part and supports computationally inexpensive adaptive BCP simulations reproducing experimentally observed shapes.
Takeaways & Limitations
The nonlinear algebraic solve may fail to converge for large time steps and creates additional convergence and error-analysis difficulties.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper introduces a Lagrange Multiplier approach for efficient, accurate, unconditionally energy-stable gradient-flow schemes. Unlike SAV-based schemes, it dissipates the original energy and does not require a lower-bounded nonlinear free-energy part, while adding a nonlinear algebraic solve.
- Gradient flows model diverse scientific and engineering processes, including crystal growth, solidification, tumor growth, thin films, phase transformations, and fracture mechanics.
- New Lagrange Multiplier approach: The new approach introduces a scalar function η(t) as a Lagrange Multiplier, with η(0)=1 ensuring equivalence to the original gradient-flow system.
- New Lagrange Multiplier approach: The continuous formulation dissipates the original energy according to dE(φ)/dt = −(µ, Gµ), unlike SAV formulations that dissipate a modified energy.
- Efficient implementation: Each time step requires two linear constant-coefficient solves and one nonlinear algebraic equation, whose cost is negligible relative to the linear solves.
- Numerical validation: The schemes achieve second-order temporal convergence for Allen-Cahn and Cahn-Hilliard tests, while computed energies decay over time.
3. Gradient flows with multiple components
The paper extends the new Lagrange multiplier approach to coupled gradient flows, proving unconditional energy stability and demonstrating second-order accuracy for coupled Cahn–Hilliard schemes.
- Model and formulation: The approach treats coupled non-local Cahn–Hilliard equations as a multi-component gradient flow with a total free energy.The model contains phase variables u and v, mobility parameters, interfacial-width parameters, coupling terms, and a nonlocal contribution.
- Stability: The resulting numerical scheme is unconditionally energy stable.The stability law is established by combining the coupled update equations with the energy-dissipation relation.
- Efficient implementation: The coupled scheme is implemented by solving four linear, constant-coefficient fourth-order equations and one nonlinear algebraic equation for ηn+1.For quartic W(u, v), the multiplier equation has degree four and is solved by Newton iteration initialized at 1.
- Accuracy validation: Both BDF2 and Crank–Nicolson achieve second-order convergence in time for the coupled model.The tests use L∞ errors at t = 0.1 for phase variables u and v with various time steps.
- Numerical validation: With Su = Sv = 1, all tested energy curves decay, while curves coincide for δt ≤ 8 × 10^-4 and differ at larger steps.The stabilized scheme permits larger accurate time steps than the unstabilized version.
4. Adaptive time stepping
The new Lagrange multiplier schemes can be combined with adaptive time stepping while retaining unconditional energy stability. Numerical tests show agreement with a fine-step reference and substantial variation in selected time steps.
- Adaptive strategy: Adaptive time stepping is combined with the new Lagrange multiplier approach to improve efficiency.The paper constructs a variable-step second-order Crank–Nicolson scheme for the coupled problem.
- Stability: The variable-step Crank–Nicolson scheme remains unconditionally energy stable.The stability result is stated for the adaptive scheme itself.
- Numerical behavior: Adaptive stepping matches the δt = 10^-5 reference energy curve, whereas uniform δt = 10^-2 deviates.The method estimates error, recalculates rejected steps, and updates accepted steps within prescribed minimum and maximum bounds.
- Numerical behavior: The largest adaptive time step is almost two orders of magnitude larger than the smallest.The associated multiplier η oscillates around 1 while remaining positive.
5. Numerical simulations of block copolymers
Two-dimensional coupled block-copolymer simulations reproduce experimentally observed morphologies across parameter regimes. Changing coupling parameters or reducing interfacial width alters domain arrangement and coarsening behavior.
- Model and physical setting: The coupled model describes macrophase separation through u and microphase separation within copolymer domains through v.The simulations use mobility, interfacial-width, temperature-related, and coupling parameters to control the dynamics.
- Computational motivation: The simulations address severe time-step constraints that make earlier semi-implicit schemes expensive, especially in three dimensions.The new stabilized scheme is used with adaptive time stepping and uniformly distributed random initial conditions.
- Annealing simulations: At t = 2, the first simulation reaches a steady state with locally striped v patterns and a morphology resembling experimental striped ellipsoids.Yellow and blue bulk regions represent A-BCP and B-BCP particles, respectively.
- Parameter effects: Reversing the sign of β preserves similar configurations but interchanges the yellow and blue bulk positions in v.This behavior appears in both the lower-temperature and higher-temperature parameter sets.
- Higher-temperature annealing: Reducing ϵv to 0.01 produces more pronounced coarsening and transformation shapes similar to experimental annealing in water.The simulations also show striped ellipsoids under the higher-temperature parameter choice.
6. Concluding remarks
The new Lagrange Multiplier approach provides efficient, unconditionally energy-stable schemes that dissipate the original energy without requiring a lower bound on the nonlinear free-energy component. Numerical studies validate the approach and show that adaptive two-dimensional coupled-BCP simulations capture experimentally observed configurations.
- The approach yields at least second-order unconditionally energy-stable schemes with computational cost similar to SAV schemes.
- Unlike SAV schemes, it dissipates the original energy and does not require the nonlinear free-energy functional to be bounded from below.
- Numerical tests found that Newton iteration typically converges in a few steps when time steps remain within the range required for accuracy.
- The stability results extend to consistent finite-dimensional Galerkin approximations because the proofs use a variational formulation with matching trial and test spaces.
- Two-dimensional adaptive simulations of the coupled BCP model captured configuration shapes observed in experiments and three-dimensional simulations.