Linear, decoupled and positivity-preserving staggered mesh schemes for general dissipative systems with arbitrary energy distributions

TL;DR

This work introduces a linear, second-order staggered-mesh framework for general nonlinear dissipative systems with arbitrary energy distributions, enabling unconditional energy stability and decoupled computations through auxiliary variables. It develops two CN-SM formulations: one for systems with known energy lower bounds using $V=E(u)$ and a positivity-preserving update, and one for unknown energy lower bounds using $V=E_{tot}(u)$ with a $\arctan$-based dissipation law, both yielding rigorous stability and convergence results. The authors provide comprehensive error analysis and confirm the theoretical results with extensive numerical experiments across Allen–Cahn, Cahn–Hilliard, Navier–Stokes, MBE, and ternary CH models, comparing favorably to LM and GSAV methods. The approach offers a robust, efficient, and broadly applicable tool for long-time simulations of dissipative systems, with strong potential for higher-order extensions and application to complex multi-physics problems.

Abstract

In this paper, we develop a novel staggered mesh (SM) approach for general nonlinear dissipative systems with arbitrary energy distributions (including cases with known or unknown energy lower bounds). Based on this framework, we propose several second-order semi-discrete schemes that maintain linearity, computational decoupling, and unconditional energy stability. Firstly, for dissipative systems with known energy lower bounds, we introduce a positive auxiliary variable $V(t)$ to substitute the total energy functional, subsequently discretizing it on staggered temporal meshes to ensure that the energy remains non-increasing regardless of the size of time step. The newly developed schemes achieve full computational decoupling, maintaining essentially the same computational expense as conventional implicit-explicit methods while demonstrating significantly improved accuracy. Furthermore, we rigorously establish the positivity preservation of the discrete variable $V^{n+1/2}$ which is a crucial property ensuring numerical stability and accuracy. Theoretical analysis confirms second-order temporal convergence for the proposed SM schemes. Secondly, for dissipative systems lacking well-defined energy lower bounds, we devise an alternative auxiliary variable formulation and extend the SM framework to maintain unconditional energy stability while preserving numerical effectiveness and accuracy. Finally, comprehensive numerical experiments, including benchmark problem simulations, validate the proposed schemes' efficacy and demonstrate their superior performance characteristics.

Figures (11)

• Figure 1: Numerical convergence rates for the Allen-Cahn equation.
• Figure 2: Numerical convergence rates for the Cahn-Hillard equation.
• Figure 3: (a): Numerical convergence rates of the Navier-Stokes equations; (b)-(c): Evolution of energy and parameter $\eta$ with $\Delta t=3 \times 10^{-4}$, $\rho=100$, $\nu=5 \times 10^{-5}$ and $\sigma=0.05$.
• Figure 4: The evolution of vorticity of Navier-Stokes equations with $\Delta t=3 \times 10^{-4}$, $\rho=100$, $\nu=5 \times 10^{-5}$, $\sigma=0.05$ at $T = 0.8, \ 1,\ 1.2$.
• Figure 5: Evolution of the vorticity at different times $t=0.35,\ 0.6,\ 1.5,\ 1.8,\ 5,\ 8$.

Theorems & Definitions (19)

• Theorem 2.1
• Proof 1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Lemma 2.8
• Lemma 2.9