Adaptively Detect and Accurately Resolve Macro-scale Shocks in an Efficient Equation-Free Multiscale Simulation

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.