A Unified Design of Energy Stable Schemes with Variable Steps for Fractional Gradient Flows and Nonlinear Integro-differential Equations
Ren‐jun Qi, Xuan Zhao
Abstract
.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 long-time 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.Keywordstime fractional gradient flownonlinear integro-differential equationnonuniform time stepsdiscrete gradient structurescalar auxiliary variableenergy stabilityMSC codes35Q9926A3365M7065M1235R11