Approximation of Scalar Conservation Law with Hysteresis
Małgorzata Peszyńska, Ralph E. Showalter · SIAM Journal on Numerical Analysis · 2020
We consider a scalar conservation law with hysteresis in which the flux function depends on the history of the evolution. The work is motivated by the need to account for hysteresis in important applications to transport with adsorption. To model hysteresis we use an auxiliary system of ODEs with constraints known as linear play combined with particularly chosen nonlinear truncation functions. A combination of these allows a quite general shape of hysteresis graph with piecewise linear primary and secondary scanning curves; this is distinct from the well-known Preisach model. We prove well-posedness of the model and stability of an explicit-implicit finite difference scheme. The challenge of history dependence requires auxiliary results in product spaces. Numerical results confirm convergence of linear or slower rate.