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The Equation-Free approach to efficient multiscale numerical computationmarries trusted micro-scale simulations to a framework for numericalmacro-scale reduction -- the patch dynamics scheme. A recent novel patch schemeempowered the Equation-Free approach to simulate systems containing shocks onthe macro-scale. However, the scheme did not predict the formation of shocksaccurately, and it could not simulate moving shocks. This article resolves bothissues, as a first step in one spatial dimension, by embedding theEquation-Free, shock-resolving patch scheme within a classic framework foradaptive moving meshes. Our canonical micro-scale problems exhibitheterogeneous nonlinear advection and heterogeneous diffusion. We demonstratemany remarkable benefits from the moving patch scheme, including efficient andaccurate macro-scale prediction despite the unknown macro-scale closure.Equation-free methods are here extended to simulate moving, forming and mergingshocks without a priori knowledge of the existence or closure of the shocks.Whereas adaptive moving mesh equations are typically stiff, typically requiringsmall time-steps on the macro-scale, the moving macro-scale mesh of patches istypically not stiff given the context of the micro-scale time-steps requiredfor the sub-patch dynamics.

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