Adaptively Detect and Accurately Resolve Macro-scale Shocks in an Efficient Equation-Free Multiscale Simulation
Date
2022
Authors
Maclean, J.
Bunder, J.E.
Kevrekidis, I.G.
Roberts, A.J.
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Journal article
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SIAM Journal on Scientific Computing, 2022; 44(4):A2557-A2581
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John Maclean, J. E. Bunder, I. G. Kevrekidis, and 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 subpatch dynamics.
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© 2022 Society for Industrial and Applied Mathematics