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Holistic projection of initial conditions onto a finite difference approximation

A. J. Roberts

TL;DR

The paper addresses how to initialize finite-difference discretisations of PDEs so that initial transients and subgrid dynamics are faithfully captured. It employs centre-manifold theory to derive a conservative initial-condition projection $u_j(0)=\langle z_j,u_0\rangle$ onto the discretisation, with projection vectors $z_j(x)$ obtained from an adjoint problem under internal boundary conditions. For Burgers' equation, this yields a discretisation that recovers standard terms while incorporating ${\cal O}(a^2)$ corrections and provides a principled method for distributing localized initial data across grid cells. The approach is extensible to higher dimensions, where subgrid problems reduce to Poisson equations with forced Neumann conditions, promising improved accuracy in short-time forecasts and subgrid fidelity.

Abstract

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of \pde{}s such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky equation (MacKenzie, 2000). In this paper the geometric view of a centre manifold is used to provide correct initial conditions for numerical discretisations (Roberts, 1997). The derived projection of initial conditions follows from the physical processes expressed in the PDEs and so is appropriately conservative. This rational approach increases the accuracy of forecasts made with finite difference models.

Holistic projection of initial conditions onto a finite difference approximation

TL;DR

The paper addresses how to initialize finite-difference discretisations of PDEs so that initial transients and subgrid dynamics are faithfully captured. It employs centre-manifold theory to derive a conservative initial-condition projection onto the discretisation, with projection vectors obtained from an adjoint problem under internal boundary conditions. For Burgers' equation, this yields a discretisation that recovers standard terms while incorporating corrections and provides a principled method for distributing localized initial data across grid cells. The approach is extensible to higher dimensions, where subgrid problems reduce to Poisson equations with forced Neumann conditions, promising improved accuracy in short-time forecasts and subgrid fidelity.

Abstract

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of \pde{}s such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky equation (MacKenzie, 2000). In this paper the geometric view of a centre manifold is used to provide correct initial conditions for numerical discretisations (Roberts, 1997). The derived projection of initial conditions follows from the physical processes expressed in the PDEs and so is appropriately conservative. This rational approach increases the accuracy of forecasts made with finite difference models.

Paper Structure

This paper contains 8 sections, 19 equations, 2 figures.

Table of Contents

  1. PACS:
  2. Keywords:
  3. Introduction
  4. Burgers' equation is discretised with centre manifold theory
  5. Project onto Burgers' discretisation
  6. Linear diffusion
  7. Nonlinear dynamics
  8. Conclusion

Figures (2)

  • Figure 1: leading two orders of approximation to the projection vectors $z_j(x)$ for purely diffusive dynamics, errors: ${\cal O}\left(\gamma\right)$, dashed; ${\cal O}\left(\gamma^2\right)$, solid.
  • Figure 2: initial fields $u=v(\hbox{\boldmath$u$}(0),x)$ corresponding to a unit-mass point release at: $\xi=0$, dot-dash; $\xi=1/4$, dashed; $\xi=1/2$, solid.