Two novel families of multiscale staggered patch schemes efficiently simulate large-scale, weakly damped, linear waves

Abstract

Many multiscale wave systems exhibit macroscale emergent behaviour, for example, the fluid dynamics of floods and tsunamis. Resolving a large range of spatial scales typically requires prohibitively high computational costs. The small dissipation in wave systems poses a significant challenge to further developing multiscale modelling methods in multiple dimensions. This article develops and evaluates two families of equation-free multiscale methods on novel 2D staggered patch schemes, and demonstrates the power and utility of these multiscale schemes for weakly damped linear waves. A detailed study of sensitivity to numerical roundoff errors establishes the robustness of developed staggered patch schemes. Comprehensive eigenvalue analysis over a wide range of parameters establishes the stability, accuracy, and consistency of the multiscale schemes. Analysis of the computational complexity shows that the measured compute times of the multiscale schemes may be 100,000 times smaller than the compute time for the corresponding (same resolution) full-domain computation. This work provides the essential foundation for efficient large-scale simulation of challenging nonlinear multiscale waves.