A UNIFIED DESIGN OF ENERGY STABLE SCHEMES WITH VARIABLE STEPS FOR FRACTIONAL GRADIENT FLOWS AND NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS.

A unified discrete gradient structure of the second order nonuniform integral averaged approximations for the Caputo fractional derivative and the Riemann--Liouville fractional integral is established in this paper. The required constraint of the step-size ratio is weaker than that found in the literature. With the proposed discrete gradient structure, the energy stability of the variable step Crank--Nicolson type numerical schemes is derived immediately, which is essential to the longtime simulations of the time fractional gradient flows and the nonlinear integro-differential models. The discrete energy dissipation laws fit seamlessly into their classical counterparts as the fractional indexes tend to one. In particular, we provide a framework for the stability analysis of variable step numerical schemes based on the scalar auxiliary variable type approaches. The time fractional Swift--Hohenberg model and the time fractional sine-Gordon model are taken as two examples to elucidate the theoretical results at great length. Extensive numerical experiments using the adaptive time-stepping strategy are provided to verify the theoretical results in the time multiscale simulations. [ABSTRACT FROM AUTHOR]