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Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method

Shu Liu, Xinzhe Zuo, Stanley Osher, Wuchen Li

TL;DR

The paper addresses efficiently solving time-implicit reaction-diffusion schemes by recasting each implicit step as a saddle-point problem with a quadratic regularization. It develops a preconditioned PDHG algorithm with explicit primal updates and FFT/DCT-based operators, achieving convergence rates that are independent of the spatial grid size. The authors prove unique solvability of the time-implicit RD scheme under mild spectral and Lipschitz conditions, and establish Lyapunov-based exponential convergence for both the continuous PDHG flow and the discrete-time algorithm, with explicit results for Allen–Cahn and Cahn–Hilliard models. Numerical experiments across multiple RD models validate grid-size independence, demonstrate effective hyperparameter selection, and show competitive efficiency against classical root-finding methods, including adaptive time stepping for long-time simulations. The work provides a practical, scalable, and theoretically justified framework for solving nonlinear RD equations at large scales.

Abstract

In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section.

Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method

TL;DR

The paper addresses efficiently solving time-implicit reaction-diffusion schemes by recasting each implicit step as a saddle-point problem with a quadratic regularization. It develops a preconditioned PDHG algorithm with explicit primal updates and FFT/DCT-based operators, achieving convergence rates that are independent of the spatial grid size. The authors prove unique solvability of the time-implicit RD scheme under mild spectral and Lipschitz conditions, and establish Lyapunov-based exponential convergence for both the continuous PDHG flow and the discrete-time algorithm, with explicit results for Allen–Cahn and Cahn–Hilliard models. Numerical experiments across multiple RD models validate grid-size independence, demonstrate effective hyperparameter selection, and show competitive efficiency against classical root-finding methods, including adaptive time stepping for long-time simulations. The work provides a practical, scalable, and theoretically justified framework for solving nonlinear RD equations at large scales.

Abstract

In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section.
Paper Structure (32 sections, 20 theorems, 196 equations, 16 figures, 3 tables)

This paper contains 32 sections, 20 theorems, 196 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Derivation of the method
  3. PDHG method for preconditioned root-finding problem
  4. Complexity of the algorithm
  5. Computing with time causality
  6. Relation with G-prox PDHG method
  7. Numerical analysis of the proposed method
  8. Unique solvability of the time-implicit scheme
  9. Lyapunov analysis for the PDHG flow
  10. Convergence analysis for the general root-finding problem
  11. Convergence analysis for our specific root-finding problem \ref{['def: preconded F']}
  12. Numerical verification
  13. Lyapunov analysis for the time-discrete case
  14. Numerical examples
  15. Tested equations
  16. ...and 17 more sections

Key Result

Theorem 1

Suppose that $\mathcal{G}_h$, $\mathcal{L}_h$ used in the finite difference scheme time-implicit scheme are self-adjoint and positive semidefinite. Assume $\mathcal{G}_h$ has the spectral decomposition as in spectral decomp Gh. We also assume that $f\in\mathcal{F}$, such that the convex function $ for some $K\geq 0.$ If the time step size $h_t$ in time-implicit scheme satisfies then the root-fi

Figures (16)

  • Figure 1: Convergence rate of the residual term $\|\widehat{F}(U_k)\|$ w.r.t. $h_t, N_t$ for Allen-Cahn equation.
  • Figure 2: Convergence rate of the residual term $\|\widehat{F}(U_k)\|$ w.r.t. $h_t, N_t$ for Cahn-Hilliard equation.
  • Figure 3: $N_{\mathrm{max}} - h_t$ log-log plot for Cahn-Hilliard equation \ref{['CH equ']}. We solve the equation on $64\time 64$ grid with $h_t = 0.01\cdot k$, $k=0.5, 1, 2, \dots, 13$. The yellow triangle has slope equals to $\frac{1}{2}$. The orange dashed line is the linear regression of data points with rather large $h_t = 0.01\cdot k$ with $5\leq k\leq 11$.
  • Figure 4: Plots of $\bar{r}$ vs $(N_t, h_t)$.
  • Figure 5: Plot of $\log\|U_k-U_*\|^2$ vs $k$ ($1\leq k \leq 400$) when using hyperparameters specified in Table \ref{['tab: theoretical rate vs actual rate']} to solve Allen-Cahn equation \ref{['AC_equ']} with different $\epsilon_0$ on a $128\times 128$ grid.
  • ...and 11 more figures

Theorems & Definitions (46)

  • Remark 1: Allen-Cahn and Cahn-Hilliard equations
  • Remark 2
  • Remark 3: Invertibility of $\mathscr{M}$
  • Theorem 1: Existence and uniqueness of \ref{['root-finding']}
  • Remark 4
  • Lemma 2: Exponential decay of $\mathcal{I}_\mu(U_t, Q_t)$
  • Theorem 3: Exponential decay of the residual $\|\widehat{F}(U_t)\|$
  • Remark 5
  • Lemma 4
  • Theorem 5: First convergence result of $\|\widehat{F}(U_t)\|$
  • ...and 36 more