Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models

by   Hong-lin Liao, et al.
Nanjing University of Aeronautics and Astronautics
Southwestern University of Finance and Economics

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 fractional Riemann-Liouville integral and 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.


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