High order numerical methods for flows with hysteretic fluxes

Abstract

A second order method and a limited variation WENO numerical scheme are proposed for the Scanning Hysteresis Model (SHM) which describes continuum flows with hysteretic looping fluxes with local memory. The SHM is a nonconservative hyperbolic system with discontinuous coefficients. The limited variation WENO method is fifth order accurate in spatial coordinate and third order accurate in time steps in smooth and monotone regions. The proposed high resolution methods produce the desired solutions more efficiently than the first order upwinding method. These methods are total variation diminishing for the flux, and monotonicity-preserving for the flux and the hysteresis state variable. The flux and the hysteresis state variable produced by these two methods satisfy maximum principle. These two methods also have invariant regions.