Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Primal-dual Forward-backward Splitting Method for Cross-diffusion Gradient Flows with General Mobility Matrices

Yunhong Deng, Chaozhen Wei

Abstract

In this work, we construct a primal-dual forward-backward (PDFB) splitting method for computing a class of cross-diffusion systems that can be formulated as gradient flows under transport distances induced by matrix mobilities. By leveraging their gradient flow structure, we use minimizing movements as the variational formulation and compute these cross-diffusion systems by solving the minimizing movements as optimization problems at the fully discrete level. Our strategy to solve the optimization problems is the PDFB splitting method outlined in our previous work \cite{PDFB2024}. The efficiency of the proposed PDFB splitting method is demonstrated on several challenging cross-diffusion equations from the literature.

A Primal-dual Forward-backward Splitting Method for Cross-diffusion Gradient Flows with General Mobility Matrices

Abstract

In this work, we construct a primal-dual forward-backward (PDFB) splitting method for computing a class of cross-diffusion systems that can be formulated as gradient flows under transport distances induced by matrix mobilities. By leveraging their gradient flow structure, we use minimizing movements as the variational formulation and compute these cross-diffusion systems by solving the minimizing movements as optimization problems at the fully discrete level. Our strategy to solve the optimization problems is the PDFB splitting method outlined in our previous work \cite{PDFB2024}. The efficiency of the proposed PDFB splitting method is demonstrated on several challenging cross-diffusion equations from the literature.

Paper Structure

This paper contains 20 sections, 5 theorems, 126 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Gradient flow structure
  3. Energy dissipation inequality
  4. Transport metric
  5. Minimizing movements
  6. Conforming approximation on staggered grids
  7. Single-step approximation
  8. Saddle point form
  9. Staggered grid approximation
  10. Construction
  11. Summary
  12. PDFB splitting optimization
  13. Algorithm
  14. Implementation details
  15. Proximal operator of primal variable
  16. ...and 5 more sections

Key Result

Lemma 2.1

\newlabelEDI-lemma0 Let $t \mapsto \boldsymbol{\mu}(t)$ be the solution of 1-1 with initial value $\boldsymbol{\mu}_{0}$, it can be equivalently characterized by where $A^{\ast}$ is the Legendre transform of $A$ in the second variable, given by where $\boldsymbol{\varphi}^{\ast}$ is the solution to the following Poisson problem:

Figures (5)

  • Figure 1: Numerical results in Example 1 in one dimension. (a) The numerical solution at several time steps, where the dash-lines represent the reference solution. (b) Plot of the relative errors at $t = 1$ with respect to different time step sizes.
  • Figure 2: Numerical results in Example 1 in two dimension. (a) The initial values of the two species, respectively. (b) Numerical solution of the two species at $t = 0.5$, respectively. (c) Numerical solution of the two species at $t = 1$, respectively. (d) The error of the numerical solution of the second species at $t = 0.5$. (e) The error of the numerical solution of the second species at $t = 1$. In (d) and (e), the error is evaluated via $|\mu_{2} - \overline{\mu}_{2}|$ where $\overline{\mu}_{2}$ is the reference solution.
  • Figure 3: Numerical results in Example 2. (a) Numerical solution at several time steps obtained with a coarser grid spacing ($h = 1/128$). (b) Numerical solution at several time steps obtained with a finer grid spacing ($h = 1/512$).
  • Figure 4: Numerical results in Example 3. (a) The numerical solution at several time steps. (b) The numerical solution computed at $t = 0.5$ which can be seen as an approximation the stationary state of the problem.
  • Figure 5: Numerical results in Example 4. (a) The plot of the initial value. (b) The evolution of the discrete free energies. (c) Stationary state of the problem (with saturation effect). (d) Stationary state of the problem without saturation, i.e., the mobility $M(\boldsymbol{\mu}) = \text{diag}(\boldsymbol{\mu})$. In (a), (c), and (d), the blue line represents the numerical solution of $\mu_{1}$, and the green color represents the line represents the saturation level, i.e., $\mu_{1} + \mu_{2}$.

Theorems & Definitions (19)

  • Definition 1.1: Time-discrete scheme
  • Lemma 2.1: EDI form of gradient flow
  • Proof 1: Proof of Lemma \ref{['EDI-lemma']}
  • Proposition 2.2
  • Proof 2
  • Lemma 2.3: Savare2016
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 9 more