[Paper Review] An O(N) quasi-Ewald splitting method for nanoconfined electrostatics

Z. Z. Gan, Xuanzhao Gao+1 more|arXiv (Cornell University)|Jan 6, 2026

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TL;DR

The paper introduces a quasi-Ewald splitting strategy (QEM) tailored for quasi-2D nanoconfined electrostatics, achieving O(N) complexity and handling dielectric interfaces without image charges.

ABSTRACT

Simulating the dynamics of charged particles in quasi-two-dimensional (quasi-2D) nanoconfined systems presents a significant computational challenge due to the long-range nature of electrostatic interactions and the geometric anisotropy. To address this, we introduce a novel quasi-Ewald splitting strategy tailored for particle-based simulations in such geometry. Our splitting strategy seamlessly integrates a collection of advanced numerical techniques, including optimal quadrature rules [L. N. Trefethen, SIAM Rev. 64(1)(2022), pp.132-150], fast pairwise kernel summation methods [S. Jiang and L. Greengard, Commun. Comput. Phys. 31(1)(2022), pp.1-26], and the random batch method with importance sampling in k-space [S. Jin, L. Li, Z. Xu et al., SIAM J. Sci. Comput. 43(4)(2021), pp.B937-B960]. The resulting algorithm achieves an O(N) overall computational complexity, where N denotes the total number of confined particles. Simulations of several prototype systems validate the accuracy and efficiency of our method. Furthermore, we present numerical observations specifically related to nanoconfined charged many-body systems, highlighting phenomena such as dielectric boundary effects, anisotropic diffusion, and the structure of the electrical double layer (EDL) under conditions of charge asymmetry.

Motivation & Objective

• Motivate efficient simulation of charged particles in quasi-two-dimensional nanoconfined geometries with dielectric interfaces.
• Develop an O(N) method that combines a quasi-Ewald splitting with specialized numerical techniques for this geometry.
• Derive and implement a Green’s function for quasi-2D nanoconfined electrostatics via Dirichlet-to-Neumann maps.
• Ensure mesh-free computation that avoids image charges while maintaining accuracy and stability.

Proposed method

• Introduce a quasi-Ewald splitting that decomposes the Coulomb kernel into short-range real-space and long-range Fourier-space components.
• Derive the Green’s function for quasi-2D nanoconfinement using a Dirichlet-to-Neumann map and provide a lattice summation form.
• Apply an optimal quadrature rule for the short-range Hankel-transformed component to achieve accurate real-space evaluation.
• Use a separation via sorting to accelerate long-range pairwise summations in Fourier space.
• Incorporate random batch sampling in k-space to obtain overall O(N) computational complexity.

Figure 3: Absolute error $\mathcal{E}$ in the evaluation of $\displaystyle\int_{0}^{\infty}\mathcal{J}_{0}(50x)e^{-x}\,\mathrm{d}x$ and $\displaystyle\int_{-\infty}^{\infty}\mathcal{J}_{0}(50x)e^{-x^{2}}\,\mathrm{d}x$ using (a). Gauss–Laguerre (half interval), and (b). Gauss–Hermite (full interval)

Experimental results

Research questions

• RQ1 How can one formulate an efficient, linear-scaling electrostatic solver for quasi-2D systems with sharp dielectric interfaces?
• RQ2 Can a quasi-Ewald splitting be constructed that naturally respects quasi-2D geometry and dielectric contrasts without image charges?
• RQ3 What numerical techniques enable accurate evaluation of the short-range and long-range components in this setting?
• RQ4 What is the expected computational complexity and accuracy of the proposed QEM in large-scale simulations?

Key findings

• The quasi-Ewald splitting yields an O(N) overall complexity for confined charged systems.
• The Green’s function for quasi-2D nanoconfinement is obtained via the Dirichlet-to-Neumann map, enabling a fast, convergent lattice sum.
• An optimal Gauss quadrature-based approach efficiently evaluates the Hankel-transformed short-range component.
• Separation via sorting boosts the efficiency of the long-range pairwise summation in k-space.
• Random batch sampling in k-space contributes to the linear scaling and facilitates MD simulations of large systems.
• The method captures dielectric boundary effects, anisotropic diffusion, and EDL structure under charge asymmetry in nanoconfined systems.

Figure 4: Required quadrature order $n$ vs. oscillation parameter $\rho\in[0,10]$ for the prescribed accuracy $\mathcal{E}$ , using truncated Gauss–Legendre quadrature for (a) $\int_{0}^{\infty}J_{0}(\rho x)e^{-x}\,\mathrm{d}x$ on $[0,k_{a}]$ ; and (b) $\int_{-\infty}^{\infty}J_{0}(\rho x)e^{-x^{2}}