Staggered grids for multidimensional multiscale modelling

TL;DR

This paper introduces multidimensional multiscale staggered grids that improve accuracy and stability in wave simulations by better preserving wave characteristics and reducing dispersion errors, especially in complex physical phenomena.

Contribution

It develops 120 multiscale staggered grid schemes for multidimensional wave modelling, demonstrating their stability, accuracy, and wave-preserving properties in equation-free multiscale contexts.

Findings

Staggered grids significantly reduce dispersion errors.

The schemes accurately preserve wave group velocity.

They demonstrate stability and improved wave characteristics in simulations.

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.