Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern

TL;DR

The paper develops a holistically discretised, amplitude-based model for spatial pattern evolution governed by the Swift-Hohenberg equation, using centre manifold theory at finite grid size $h$ to derive robust discrete dynamics for amplitudes $a_j$ (and $b_j$). It introduces inter-element coupling via a parameter $\gamma$ and computes the evolution equations, e.g. $\dot a_j = r a_j + \frac{4\gamma^2}{h^2}\delta^2 a_j - 3\gamma^2 a_j^2 b_j + \mathcal{O}(\gamma^4 + A^4 + r^2)$ and a corresponding equation for $\dot b_j$, with $\gamma=1$ recovering the discrete Ginzburg-Landau type dynamics. The method explicitly handles boundary conditions by modifying interfacial boundary conditions, producing boundary-layer subgrid structure that enforces prescribed boundary data and captures boundary forcing effects. The approach generalises to higher dimensions and higher-order expansions via computer algebra, offering a principled route to accurate, stable pattern dynamics and boundary influence modelling beyond the specific Swift-Hohenberg system.

Abstract

I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the pattern amplitude, has the rigorous support of centre manifold theory at finite grid size $h$, and naturally incorporates physical boundaries. The results presented here for the Swift-Hohenberg equation suggest the approach will form a powerful method in computationally exploring pattern selection in general. With the aid of computer algebra, the techniques may be applied to a wide variety of equations to derive numerical models that accurately and stably capture the dynamics including the influence of possibly forced boundaries.

Figures (2)

• Figure 1: schematics diagram showing a varying "roll" structure (solid curve) discretised by introducing artificial internal boundaries every period ($p=1$) at odd multiples of $\pi$ (vertical bars). The "roll" field in the $j$th element is parametrised by the amplitude $a_j$ (discs).
• Figure 2: spatial subgrid structure (\ref{['eq:fld1']}) of the coefficients of $\alpha$ (solid), $\beta$ (dashed) and its second derivative (dotted) when the boundary condition at $x=x_1-\pi$ is (\ref{['eq:evenbc']}) for an element size $h=2\pi$ . These give the field if all amplitudes $a_j=b_j=0$ . See the $\alpha$ and the second derivative curves are at the boundary $x=0$ about a value of one higher than the field in the bulk of the element---this is appropriate as $\alpha$ specifies the boundary value and $\beta$ the second derivative.