Staggered grids for multidimensional multiscale modelling

Abstract

For high accuracy and to improve simulated wave characteristics, this article extends the concept of staggered grids to novel multidimensional multiscale modelling enabling efficient computation on sparse patches. Computational schemes for wave-like systems with small dissipation are often inaccurate and unstable due to truncation errors and numerical roundoff errors. Hence 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. But numerical schemes on staggered grids are significantly less dispersive, better model the group velocity, and preserve much of the wave characteristics. This article develops and exhaustively studies all 167040 possible 2D multiscale staggered grids. Our catalog (Divahar, 2023) interactively plots all of them. Only 120 multiscale staggered patch grids give stable and accurate multiscale schemes. Specifically, this article develops these 120 multiscale staggered grids and demonstrates their stability, accuracy, and wave-preserving characteristic for equation-free multiscale modelling of weakly damped linear waves. These 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.