Discrete Gradient Structure of a Second-Order Variable-Step Method for Nonlinear Integro-Differential Models

.The discrete gradient structure and the positive definiteness of discrete fractional integrals or derivatives are fundamental to the numerical stability in long-time simulation of nonlinear integro-differential models. We build up a discrete gradient structure for a class of second-order variable-step approximations of the fractional Riemann–Liouville integral and the fractional Caputo derivative. Then certain variational energy dissipation laws at discrete levels of the corresponding variable-step Crank–Nicolson-type methods are established for time-fractional Allen–Cahn and time-fractional Klein–Gordon-type models. They are shown to be asymptotically compatible with the associated energy laws of the classical Allen–Cahn and Klein–Gordon equations in the associated fractional order limits. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of our second-order methods.Keywordsintegral averaged formuladiscrete gradient structuretime-fractional Allen–Cahn modeltime-fractional Klein–Gordon modeldiscrete variational energy lawMSC codes35Q9965M0665M1274A50