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

TL;DR

This paper introduces an adaptive moving patch scheme within the Equation-Free framework to accurately and efficiently simulate macro-scale shocks, including moving and forming shocks, without prior shock knowledge.

Contribution

It integrates adaptive moving meshes into the Equation-Free patch scheme, enabling accurate simulation of dynamic shocks in multiscale systems without prior shock information.

Findings

Successfully simulates moving and forming shocks

Achieves accurate macro-scale predictions with efficient computation

Extends Equation-Free methods to complex shock dynamics

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.