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Discrete versus continuous -- lattice models and their exact continuous counterparts

Lorenzo Fusi, Oliver Křenek, Vít Průša, Casey Rodriguez, Rebecca Tozzi, Martin Vejvoda

TL;DR

The paper addresses the problem of connecting discrete lattice models with their continuous PDE counterparts through discretisation and continualisation, using Fourier analysis as the unifying toolkit. It develops a systematic framework based on bandwidth-limited interpolants and semidiscrete Fourier transforms to translate between discrete spectra and continuous dispersion relations, proving exact correspondences in infinite and periodic settings and extending to finite Dirichlet boundaries. The main contributions include explicit dispersion-relations for nearest-neighbour and general interactions, a reconstruction procedure that yields exact equivalences, and a demonstration that sine-transform-based discretisations can preserve key spectral properties even for Sturm–Liouville-type operators. The work provides principled guidance for designing dispersion-preserving discretisations and continualisation strategies, with potential impact on wave-propagation modeling and numerical analysis of lattice-continuum transitions.

Abstract

We review and study the correspondence between discrete lattice/chain models of interacting particles and their continuous counterparts represented by partial differential equations. We study the correspondence problem for nearest neighbour interaction lattice models as well as for multiple-neighbour interaction lattice models, and we gradually proceed from infinite lattices to periodic lattices and finally to finite lattices with fixed ends/zero Dirichlet boundary conditions. The whole study is framed as systematic specialisation of Fourier analysis tools from the continuous to the discrete setting and vice versa, and the correspondence between the discrete and continuous models is examined primarily with regard to the dispersion relation.

Discrete versus continuous -- lattice models and their exact continuous counterparts

TL;DR

The paper addresses the problem of connecting discrete lattice models with their continuous PDE counterparts through discretisation and continualisation, using Fourier analysis as the unifying toolkit. It develops a systematic framework based on bandwidth-limited interpolants and semidiscrete Fourier transforms to translate between discrete spectra and continuous dispersion relations, proving exact correspondences in infinite and periodic settings and extending to finite Dirichlet boundaries. The main contributions include explicit dispersion-relations for nearest-neighbour and general interactions, a reconstruction procedure that yields exact equivalences, and a demonstration that sine-transform-based discretisations can preserve key spectral properties even for Sturm–Liouville-type operators. The work provides principled guidance for designing dispersion-preserving discretisations and continualisation strategies, with potential impact on wave-propagation modeling and numerical analysis of lattice-continuum transitions.

Abstract

We review and study the correspondence between discrete lattice/chain models of interacting particles and their continuous counterparts represented by partial differential equations. We study the correspondence problem for nearest neighbour interaction lattice models as well as for multiple-neighbour interaction lattice models, and we gradually proceed from infinite lattices to periodic lattices and finally to finite lattices with fixed ends/zero Dirichlet boundary conditions. The whole study is framed as systematic specialisation of Fourier analysis tools from the continuous to the discrete setting and vice versa, and the correspondence between the discrete and continuous models is examined primarily with regard to the dispersion relation.
Paper Structure (38 sections, 8 theorems, 323 equations, 6 figures, 3 tables)

This paper contains 38 sections, 8 theorems, 323 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A further look into discretisation and continualisation problems
  3. Dispersion relation and wave propagation
  4. Eigenvalues approximation
  5. Fourier transform
  6. Infinite lattice
  7. Infinite lattice model of interacting particles
  8. Search for continuous analogue of lattice model---case study for the nearest neighbour interaction lattice model
  9. Reconstruction procedure in detail---from discrete grid values to a function of real variable via bandwidth limited interpolant
  10. Semidiscrete Fourier transform and bandwidth limited interpolant
  11. Semidiscrete Fourier transform and discrete convolution
  12. Semidiscrete Fourier transform diagonalises infinite circulant matrices/Laurent operators
  13. Semidiscrete Fourier transform and differentiation
  14. Search for continuous analogue of lattice model---revisiting the nearest neighbour interaction lattice model
  15. Correspondence between discrete and continuous models
  16. ...and 23 more sections

Key Result

theorem 1

Let $h>0$, and let $\left\{x_{h, j}\right\}_{j=-\infty}^{+\infty}$, $x_{h, j} =_{\mathrm{def}} jh$, be the corresponding grid on the real line ${\mathbb R}$. Let $\bm{u}_h(t) = \left\{ u_{h, j} (t)\right\}_{j=-\infty}^{+\infty}$ be a set of grid values on the grid $\left\{x_{h, j} \right\}_{j=-\inft Let the grid values $\bm{u}_h(t) = \left\{ u_{h, j} (t)\right\}_{j=-\infty}^{+\infty}$ solve the i

Figures (6)

  • Figure 1: Simple lattice model.
  • Figure 2: Construction of a function $u_h(t, x)$ out of grid values.
  • Figure 3: Semidiscrete Fourier transform and construction of bandwidth limited interpolant $u_h$ of grid values vector $\bm{u}_h$, equispaced grid $x_{h, j} = jh$, $j \in {\mathbb Z}$.
  • Figure 4: Function $\frac{1}{h}U_{\mathrm{Triangle}} \left( \frac{x}{h} \right)$.
  • Figure 5: Lattices.
  • ...and 1 more figures

Theorems & Definitions (14)

  • theorem 1: Equivalence between a discrete system of ordinary differential equations for grid values and the corresponding partial differential equation---nearest neighbour interaction, infinite lattice
  • theorem 2: Equivalence between discrete system of ordinary differential equations for grid values and the corresponding partial differential equation---general interaction, infinite lattice
  • proof
  • corollary 3: Exact lattice model for the standard wave equation
  • lemma 4: Eigenvalues of circulant matrices $\mathbb{C}_{\left(2M + 2\right) \times \left(2M + 2\right)}$
  • proof
  • theorem 5: Equivalence between discrete system of ordinary differential equations for grid values and the corresponding partial differential equation---general interaction, periodic lattice
  • proof
  • theorem 6: Equivalence between discrete system of ordinary differential equations for grid values and the corresponding partial differential equation---nearest neighbour interaction, fixed ends/zero Dirichlet boundary condition
  • proof
  • ...and 4 more