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arXiv · 2301.12474

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

Hong-lin Liao·Nan Liu·Pin Lyu
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Abstract

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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Hong-lin Liao, Nan Liu, Pin Lyu. 2023-01-29. Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models. https://arxiv.org/abs/2301.12474

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