This is the first talk on the joint work with Ralph Showalter. This talk will focus on the modeling and computational aspects, with a surprising connection to diophantine approximations.
Hysteresis is the dependence of a function not just on the present argument, but also on its history. In mathematical modeling, we call hysteresis the rate independent memory. Hysteresis is well-known in ferromagnetism and was first studied by Maxwell. Other applications in which hysteresis is important include capillary pressure and adsorption. This talk will focus on a scalar conservation law for transport with adsorption hysteresis. To model hysteresis we use an auxiliary system of ODEs with constraints, known as linear play, as well as nonlinear truncation functions. This latter element which we recently incorporated allows a quite general shape of hysteresis graph with piecewise linear sides. These sides are calibrated using diophantine approximations.
Our main result is the 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.