Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances

Jose A. Carrillo, Li Wang, Chaozhen Wei

TL;DR

This work develops structure-preserving numerical schemes for nonlinear mobility gradient flows, casting the evolution as a Wasserstein-like gradient flow and solving it via a generalized JKO minimization that preserves positivity, mass, and energy dissipation. A primal-dual framework (PD3O) and a fast preconditioned variant (PrePD3O) are designed to efficiently handle the nonlinear mobility and wall-boundary terms, including Newton-based proximal computations for the mobility term. The discretizations are implemented in 1D and 2D, with detailed treatment of wetting boundary conditions and energy gradients, yielding robust performance across saturated, Cahn–Hilliard, and wetting scenarios. Numerical results demonstrate second-order spatial accuracy, steady-energy decay, and significantly accelerated convergence, validating the method’s applicability to a broad class of gradient-flow problems with degenerate mobilities.

Abstract

We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On one hand, the essential properties of the solution, including positivity, global bounds, mass conservation and energy dissipation are all guaranteed by construction. On the other hand, it enjoys sufficient flexibility when applies to a large variety of problems including different free energy functionals, general wetting boundary conditions and degenerate mobilities. The performance of our methods are demonstrated through a suite of examples.

Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances

TL;DR

This work develops structure-preserving numerical schemes for nonlinear mobility gradient flows, casting the evolution as a Wasserstein-like gradient flow and solving it via a generalized JKO minimization that preserves positivity, mass, and energy dissipation. A primal-dual framework (PD3O) and a fast preconditioned variant (PrePD3O) are designed to efficiently handle the nonlinear mobility and wall-boundary terms, including Newton-based proximal computations for the mobility term. The discretizations are implemented in 1D and 2D, with detailed treatment of wetting boundary conditions and energy gradients, yielding robust performance across saturated, Cahn–Hilliard, and wetting scenarios. Numerical results demonstrate second-order spatial accuracy, steady-energy decay, and significantly accelerated convergence, validating the method’s applicability to a broad class of gradient-flow problems with degenerate mobilities.

Abstract

We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On one hand, the essential properties of the solution, including positivity, global bounds, mass conservation and energy dissipation are all guaranteed by construction. On the other hand, it enjoys sufficient flexibility when applies to a large variety of problems including different free energy functionals, general wetting boundary conditions and degenerate mobilities. The performance of our methods are demonstrated through a suite of examples.
Paper Structure (15 sections, 1 theorem, 61 equations, 12 figures, 2 algorithms)

This paper contains 15 sections, 1 theorem, 61 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Variational formulation
  3. Semi-discrete JKO scheme
  4. Fully discrete schemes
  5. One-dimensional case
  6. Two-dimensional case
  7. Primal-Dual algorithm
  8. Primal-Dual method for three operators
  9. Accelerated Primal-Dual method by preconditioning
  10. Computing $\text{Prox}_{\lambda \Phi} (u)$
  11. Numerical Results
  12. 1D Saturation Experiment
  13. 1D Cahn-Hilliard equation
  14. 2D Cahn-Hilliard equation
  15. Wetting phenomenon of droplets

Key Result

Proposition 1

The variational formulation has the following properties for any $k \geq 0$:

Figures (12)

  • Figure 1: Evolution of solutions to 1D Saturation equation with mobility $M(\rho)=\rho(1-\rho)$ for $t\in[0,15]$. Left: $\Delta x=0.04$, $\tau=0.01$; Right: $\Delta x=0.02$, $\tau=0.01$. The insets are the zoom-in figures for the oscillation when $\rho$ is close to 1.
  • Figure 2: Left: Evolution of solutions for 1D Saturation experiment by Generalized Schrödinger bridge scheme with $\Delta x=0.04$, $\tau=0.01$, $\eta_0$=80; Right: Comparison with Generalized dynamic JKO scheme.
  • Figure 3: Evolution for the 1D Cahn-Hilliard equation with logarithmic potential $H_{log}=(1-\rho^2)/2$. Left: evolution of $\rho(x,t)$. $\Delta x=0.01$, $\tau=0.001$. Center: Free energy decay. Right: Convergence to exact steady solution for various $\Delta x$.
  • Figure 4: Evolution for the 1D Cahn-Hilliard equation with logarithmic potential, where the parameters are $\theta=0.3$, $\theta_c=1$ and $\epsilon=\sqrt{10^{-3}}$. Left: evolution of $\rho(x,t)$. $\Delta x=0.0125$, $\tau=0.1$. Right: Free energy decay.
  • Figure 5: Evolution for the 1D Cahn-Hilliard equation with logarithmic potential ($\theta=0.3$, $\theta_c=1$) (top row) and double-well potential (bottom row), and $\epsilon=1$. Left: evolution of $\rho(x,t)$. $\Delta x=0.4$, $\tau=0.01$. Center: Zoom in phase-field solution at $t=100$. Right: Free energy decay.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2: Use of mix boudnary conditions
  • Remark 3: Determination of boundary values using wetting boundary conditions
  • Remark 4: Generalized Schrödinger bridge problem