Large‐scale simulation of shallow water waves via computation only on small staggered patches

• Published in: International journal for numerical methods in fluids 2021-04, Vol.93 (4), p.953-977
• Main Authors: Bunder, Judith E., Divahar, Jayaraman, Kevrekidis, Ioannis G., Mattner, Trent W., Roberts, Anthony J.
• Format: Article
• Language: English
• Subjects: Computation; Computer applications; Computing time; Coupling; Empowerment; finite difference; Interpolation; model reduction; nonlinear dynamics; partial differential equations; Patches (structures); Physical simulation; Physics; reduced‐order modeling; Shallow water; Shallow water waves; Simulation; spectral; Spectral analysis; Spectrum analysis; Water waves
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4915

A multiscale computational scheme is developed to use given small microscale simulations of complicated physical wave processes to empower macroscale system‐level predictions. By coupling small patches of simulations over unsimulated space, large savings in computational time are realizable. Here, we generalize the patch scheme to the case of wave systems on staggered grids in two‐dimensional (2D) space. Classic macroscale interpolation provides a generic coupling between patches that achieves consistency between the emergent macroscale simulation and the underlying microscale dynamics. Spectral analysis indicates that the resultant scheme empowers feasible computation of large macroscale simulations of wave systems even with complicated underlying physics. As an example of the scheme's application, we use it to simulate some simple scenarios of a given turbulent shallow water model. To accurately and efficiently simulate waves with complex microscale physics, our scheme computes on only small well‐separated patches of space. The patches are craftily coupled to ensure macroscale predictions are of proven accuracy. A macroscale staggered grid of patches, each with a staggered microgrid, ensures good wave properties. The ideal wave PDE in 2D establishes that the scheme applies to a wide range of waves in multi‐D space. Applied to a nonlinear Smagorinski‐based model of turbulent shallow water flow, the scheme demonstrates good macroscale predictions.