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arXiv · 2108.11568

Adaptively detect and accurately resolve macro-scale shocks in an efficient Equation-Free multiscale simulation

John Maclean·J. E. Bunder·I. G. Kevrekidis·A. J. Roberts
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Abstract

The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the sub-patch dynamics.

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John Maclean, J. E. Bunder, I. G. Kevrekidis, A. J. Roberts. 2021-08-26. Adaptively detect and accurately resolve macro-scale shocks in an efficient Equation-Free multiscale simulation. https://arxiv.org/abs/2108.11568

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