Accurate multiscale simulation of wave-like systems

Abstract

Many important phenomena exhibit multiscale emergent behaviour that currently cannot be accurately modelled (e.g., many turbulent flows, floods, tsunamis, weather, sediment transport) due to the high computational effort to capture all the details spread over a large range of scales. To address this issue, many multiscale modelling techniques have been developed for dissipative systems. The flexible equation-free multiscale patch schemes accurately model emergent macroscale spatial dynamics using a given microscale model within small sparsely located coupled patches. Such multiscale patch schemes have been developed successfully for dissipative systems. But the small dissipation in wave-like systems poses significant challenges for developing multiscale patch schemes, especially in multiple dimensions. The recent works of Cao and Roberts (2013, 2015) extend the patch scheme to 1D wave-like systems. This PhD thesis develops the equationfree multiscale patch schemes for 2D wave-like systems (small dissipation) and explores the schemes more thoroughly. For high accuracy and to preserve much of the wave characteristics, we extend the concept of staggered grids in the full-domain modelling to multiscale modelling. In contrast to the usual collocated patch grid, using staggered grids within the patches leads to many different arrangements of the patch edge nodes. We considered 83 520 staggered patch grids that are geometrically compatible for 2D wave-like systems. Almost all such staggered grids lead to unstable and/or inaccurate patch schemes. We designed two staggered patch grids that constitute stable and accurate patch schemes for linear and nonlinear wave-like systems. Patch coupling provides a two-way connection between two scales: from microscale within patches to macroscale over the domain, and from macroscale to microscale. Depending upon the patch coupling methods, many variants of patch schemes are possible on a staggered patch grid; many such patch schemes are unstable and/or inaccurate. We developed two novel families of equation-free multiscale staggered patch schemes for accurate large-scale simulation of wave-like systems: a spectral patch scheme, and four polynomial patch schemes. The spectral patch scheme is the best for accuracy, over simple geometry and periodic boundary conditions. Polynomial patch schemes are best for complex geometry and boundary conditions. We show that the staggered patch schemes accurately simulate general linear waves, viscous shallow water flows, and turbulent shallow water flows. For these three wave models, we establish the stability and consistency of the staggered patch schemes for a wide range of physical and grid parameters via analytic eigenvalue analysis and numerical von Neumann analysis. The eigenvalues of the patch scheme macroscale modes converge towards the corresponding eigenvalues of the full-domain model, with decreasing macro-grid interval (inter-patch distance). The staggered patch schemes are not sensitive to numerical roundoff errors, except when the patches are too small relative to inter-patch distance, and/or when the underlying microscale model is sensitive to numerical roundoff errors. Even for very small patches, we confirm the consistency of the patch schemes via arbitrary-precision floating-point implementation. We also show the accuracy of the patch scheme time simulations by comparing them with the full-domain simulation. Our work shows the robustness of the patch scheme simulation by recovering emergent macroscale waves from random initial perturbations. Explicit analytic expressions quantify the computational complexity of the staggered patch schemes. The staggered patch schemes accurately model the macroscale waves with large computational savings, via detailed microscale simulations only within the patches, that is, within a small fraction of the whole space. The staggered patch schemes compute only for a small number of dynamical state variables, as small as one-millionth of the number of state variables in the corresponding full-domain model. The measured compute times of the multiscale staggered patch schemes are up to 105 times smaller than the corresponding full-domain model. Users can choose how much computational savings to achieve depending on the scales of interest in the modelling. The patch schemes’ ability to accurately model macroscale waves with large computational savings is an enabling feature for accurate simulation and prediction of large-scale waves like floods and tsunamis.