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Stability and error analysis of pressure-correction scheme for the Navier-Stokes-Planck-Nernst-Poisson equations

Yuyu He, Hongtao Chen

TL;DR

The paper addresses the NSPNP system by developing a first-order SAV-based pressure-correction scheme that is fully decoupled and linearized, while preserving non-negativity and mass conservation and ensuring unconditional energy stability. It introduces an auxiliary variable $r(t)$ tied to the energy $E( abla heta)$ to balance nonlinear Navier–Stokes terms, enabling stability without a time-step restriction. A rigorous two-dimensional error analysis is provided via induction, giving $L^2$ and $H^1$-norm error bounds for $c_1,c_2, ablaoldsymbol{u}, abla p$ and $r$, and $H^2$ bounds for the electric potential, with optimal pressure estimates. Numerical experiments corroborate theoretical results, confirming first-order temporal convergence, energy dissipation, mass conservation, and positivity, while highlighting the limitation to 2D in the current analysis and the plan for 3D extensions.

Abstract

In this paper, we propose and analyze first-order time-stepping pressure-correction projection scheme for the Navier-Stokes-Planck-Nernst-Poisson equations. By introducing a governing equation for the auxiliary variable through the ionic concentration equations, we reconstruct the original equations into an equivalent system and develop a first-order decoupled and linearized scheme. This scheme preserves non-negativity and mass conservation of the concentration components and is unconditionally energy stable. We derive the rigorous error estimates in the two dimensional case for the ionic concentrations, electric potential, velocity and pressure in the $L^2$- and $H^1$-norms. Numerical examples are presented to validate the proposed scheme.

Stability and error analysis of pressure-correction scheme for the Navier-Stokes-Planck-Nernst-Poisson equations

TL;DR

The paper addresses the NSPNP system by developing a first-order SAV-based pressure-correction scheme that is fully decoupled and linearized, while preserving non-negativity and mass conservation and ensuring unconditional energy stability. It introduces an auxiliary variable tied to the energy to balance nonlinear Navier–Stokes terms, enabling stability without a time-step restriction. A rigorous two-dimensional error analysis is provided via induction, giving and -norm error bounds for and , and bounds for the electric potential, with optimal pressure estimates. Numerical experiments corroborate theoretical results, confirming first-order temporal convergence, energy dissipation, mass conservation, and positivity, while highlighting the limitation to 2D in the current analysis and the plan for 3D extensions.

Abstract

In this paper, we propose and analyze first-order time-stepping pressure-correction projection scheme for the Navier-Stokes-Planck-Nernst-Poisson equations. By introducing a governing equation for the auxiliary variable through the ionic concentration equations, we reconstruct the original equations into an equivalent system and develop a first-order decoupled and linearized scheme. This scheme preserves non-negativity and mass conservation of the concentration components and is unconditionally energy stable. We derive the rigorous error estimates in the two dimensional case for the ionic concentrations, electric potential, velocity and pressure in the - and -norms. Numerical examples are presented to validate the proposed scheme.
Paper Structure (9 sections, 11 theorems, 163 equations, 3 figures, 4 tables)

This paper contains 9 sections, 11 theorems, 163 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The SAV pressure-correction scheme and its stability
  3. Error analysis
  4. Estimates for the ionic concentration and potential
  5. Error estimates for the ionic concentration, potential and velocity
  6. Proof of Theorem \ref{['th3.3']}
  7. Error estimates for the pressure
  8. Numerical experiments
  9. Concluding remarks

Key Result

Theorem 2.1

\newlabelth2.10 Give the initial values $c^0_1(\bm{x})$, $c^0_2(\bm{x})$ which are non-negative almost everywhere in $\Omega$, then the scheme hy2.8 has the following properties for all $0 \leq n \leq N-1$:

Figures (3)

  • Figure 1: Time evolutions of the discrete energy and original energy for Example 3.
  • Figure 2: Time evolutions of masses $\int_{\Omega}c_1d\bm{x}$, $\int_{\Omega}c_2d\bm{x}$ and their errors with $\tau=0.005$ for Example 3.
  • Figure 3: Time evolutions of maximum and minimum values of $c_1$, $c_2$ with $\tau=0.005$ for Example 3.

Theorems & Definitions (20)

  • Theorem 2.1
  • Proof 1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Proof 2
  • ...and 10 more