A holistic finite difference approach models linear dynamics consistently

TL;DR

This paper develops a holistic finite difference framework for linear PDEs by applying centre manifold theory to a discretised, inter-element coupled system. By splitting the dynamics into an even operator $\mathcal{A}$ and an odd perturbation $\epsilon\mathcal{B}$ and introducing artificial boundary conditions governed by a coupling parameter $\gamma$, it derives low-dimensional, stencil-based models $\\dot u_j=g({\boldsymbol u})$ that accurately approximate the original PDE as both $h\to0$ and $\gamma,\epsilon\to0$, with exponential attraction to the centre manifold. The author demonstrates robust modelling for advection-diffusion, obtaining stable, upwind-like schemes at finite $\epsilon h$ and showing higher-order corrections yield $O(h^{2\ell})$-level consistency for even terms and $O(h^{2\ell-1})$ plus $O(\epsilon^2)$ for odd perturbations. These results imply that fewer grid points may suffice without sacrificing accuracy or stability, even for large advection speeds, and provide a systematic route to extend to nonlinear PDEs via centre-manifold analysis. The work combines rigorous dynamical-systems theory with computer-algebra-assisted derivations to produce numerically faithful discretisations that preserve the long-term dynamics on finite grids, with clear implications for efficient, robust numerical solvers.

Abstract

I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing. For example, the advection-diffusion equation is found to be stably modelled for all advection speeds and all grid spacing. The theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory. When numerically solving nonlinear partial differential equations, this approach can be used to derive systematically finite difference models which automatically have excellent characteristics. Their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems.

Figures (3)

• Figure 1: conceptual diagram showing: the traditional finite difference modelling approaches (rightward-arrows) the physical problem (upper disc) via asymptotic consistency as the grid size $h\to 0$ (left circles); whereas the holistic method approaches (forward-arrows) the physical problem via asymptotics in nonlinearity and the inter-element coupling $\gamma$ (from right circle). Herein we establish how to use the holistic approach to do both in order to model a general linear problem (lower disc).
• Figure 2: schematic picture of the equi-spaced grid, $x_j$ spacing $h$, the unknowns $u_j$, the artificial internal boundaries between each element (vertical lines), and in the neighbourhood of $x_j$ the field $v_j(x,t)$ which extends outside the element and, if $\gamma=1$, passes through the neighbouring grid values $u_{j\pm1}$.
• Figure 3: coefficients of the centre manifold models (\ref{['eq:hiad']}) and (\ref{['eq:hiiad']}) as a function of advection speed and grid spacing, $\epsilon h$. These curves are at least of graphical accuracy and are obtained via the Shanks transform of the Taylor series to errors ${\cal O}\left(\epsilon^{14}\right)$. The dotted lines are the presumed large $\epsilon h$ asymptotes: $\nu_1\approx \frac{\epsilon h}{2}$ ; $\nu_2\approx \frac{\epsilon h}{4}-\frac{1}{2}$ ; $\kappa_2\approx \frac{1}{2}-\frac{1}{\epsilon h}$ .