Staggered grids for multidimensional multiscale modelling

J. Divahar School of Mathematical Sciences, University of Adelaide, South Australia.\urlhttps://orcid.org/0000-0002-9506-8846 A. J. Roberts ¹¹footnotemark: 1 \urlhttp://orcid.org/0000-0001-8930-1552, \urlmailto:profajroberts@protonmail.com Trent W. Mattner ¹¹footnotemark: 1 \urlhttps://orcid.org/0000-0002-5313-5887 J. E. Bunder ¹¹footnotemark: 1 \urlhttp://orcid.org/0000-0001-5355-2288 Ioannis G. Kevrekidis Departments of Chemical and Biomolecular Engineering & Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland, USA. \urlhttps://orcid.org/0000-0003-2220-3522

Abstract

Numerical schemes for wave-like systems with small dissipation are often inaccurate and unstable due to truncation errors and numerical roundoff errors. Hence, numerical simulations of wave-like systems lacking proper handling of these numerical issues often fail to represent the physical characteristics of wave phenomena. This challenge gets even more intricate for multiscale modelling, especially in multiple dimensions. When using the usual collocated grid, about two-thirds of the resolved wave modes are incorrect with significant dispersion. But, numerical schemes on staggered grids (with alternating variable arrangement) are significantly less dispersive and preserve much of the wave characteristics. Also, the group velocity of the energy propagation in the numerical waves on a staggered grid is in the correct direction, in contrast to the collocated grid. For high accuracy and to preserve much of the wave characteristics, this article extends the concept of staggered grids in full-domain modelling to multidimensional multiscale modelling. Specifically, this article develops $120$ multiscale staggered grids and demonstrates their stability, accuracy, and wave-preserving characteristic for equation-free multiscale modelling of weakly damped linear waves. But most characteristics of the developed multiscale staggered grids must also hold in general for multiscale modelling of many complex spatio-temporal physical phenomena such as the general computational fluid dynamics.

1 Introduction

For wave-like systems with small or no dissipation, accurate numerical simulation is challenging over large spatial scales, especially for long simulation times. Numerical schemes for wave-like systems with small dissipation are often inaccurate and unstable due to numerical dissipation and numerical dispersion caused by truncation and numerical roundoff errors (\cites[p.136]Hinch2020_ThnkBfreYuCmpte_APrldeToCmpttnlFldDynmcs[pp.70–73]Zikanov2010_EsntlCmpttnlFldDynmcs[pp.232–243]Anderson1995_CmpttnlFldDynmcs). Hence, numerical simulations of wave-like systems lacking proper handling of these numerical issues often fail to represent the physical characteristics of wave phenomena. LABEL:sec:cllctdAndStgrdGrdsFrWvelkeSystms overviews using a staggered grid in space for accurate and robust computational simulation of wave-like systems. It is well-known that staggered grids, such as depicted in LABEL:fig:flDmnGrd_clctd_stgrd(right), lead to higher accuracy compared to that of a same order scheme on collocated grids.

Here we develop multiscale staggered grids for wave-like systems in multiple space dimensions—such systems are even more challenging (e.g., see the recent review of modelling materials by