Adaptive time-stepping and maximum-principle preserving Lagrangian schemes for gradient flows

TL;DR

The paper addresses efficiently simulating gradient flows with both non-conservative and conservative dynamics by introducing adaptive, second-order Lagrangian schemes. It develops a flow-map-based non-conservative formulation with determinant-positivity regularization and a Wasserstein-based conservative formulation with mass conservation, each accompanied by an adaptive BDF2 time-stepping method. Key contributions include rigorous energy-dissipation proofs and MBP/positivity results under explicit time-step ratio bounds, along with comprehensive 1D and 2D numerical validations for Allen-Cahn, PME, and Keller-Segel models. The methods enhance interface tracking and stability in gradient-flow simulations, offering practical benefits for free-boundary and aggregation-diffusion problems with variable time stepping.

Abstract

We develop in this paper an adaptive time-stepping approach for gradient flows with distinct treatments for conservative and non-conservative dynamics. For the non-conservative gradient flows in Lagrangian coordinates, we propose a modified formulation augmented by auxiliary terms to guarantee positivity of the determinant, and prove that the corresponding adaptive second-order Backward Difference Formulas (BDF2) scheme preserves energy stability and the maximum principle under the time-step ratio constraint $0<r_n\le r_{\max}\le\frac{3}{2}$. On the other hand, for the conservative Wasserstein gradient flows in Lagrangian coordinates, we propose an adaptive BDF2 scheme which is shown to be energy dissipative, and positivity preserving under the time-step ratio constraint $0<r_n\le r_{\max}\le\frac{3+\sqrt{17}}{2}$ in 1D and $0<r_n\le r_{\max}\le \frac{5}{4}$ in 2D, respectively. We also present ample numerical simulations in 1D and 2D to validate the efficiency and accuracy of the proposed schemes.

Figures (11)

• Figure 1: A schematic illustration of a flow map $\bm x(\bm X,t)$ at a fixed time $t$: $\bm x(\bm X,t)$ maps $\Omega_0^{\bm X}$ to $\Omega_t^{\bm x}$. $\bm X$ is the Lagrangian coordinate while $\bm x$ is the Eulerian coordinate, and $F(\bm X,t)=\frac{\partial \bm x(\bm X,t)}{\partial \bm X}$ represents the deformation associated with the flow map.
• Figure 2: Convergence order for \ref{['scheme:ac fully discrete']} with $\rho_0(X)=1-X^2$, $X\in[-1,1]$, $\epsilon=0.01$, $\eta=0$, $T=0.5$.
• Figure 3: Numerical solution solved by scheme \ref{['scheme:ac fully discrete']} at $T=20$ with $\epsilon=0.01$, $M_x=100$, $N=2000$. (a-f): $\tau_n=\frac{\sigma_nT}{\sum_{k=1}^{N}\sigma_k}$. (g-h): strategy \ref{['strategy2']} with $\beta=1e5$, $r_{\text{user}}=1.5$, $\tau_{\max}=0.1$ and $\tau_{\min}=1e-3$.
• Figure 4: Numerical results solved by scheme \ref{['eq:scheme bdf2']}-\ref{['schem:1-3']} using the Algorithm \ref{['algo,2']}. Initial value \ref{['initial:wt']} with $m=2$, $\theta=0.25$, $M_x=800$.
• Figure 5: Numerical results for PME solved by \ref{['eq:scheme bdf2']}-\ref{['schem:1-3']} with \ref{['initial:wt']}, $\theta=0.25$, $M_x=800$, $\delta t=\frac{1}{800}$.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• proof
• Theorem 3.2