Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models
| Autor: | Liao, Hong-lin, Liu, Nan, Lyu, Pin |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | SIAM Journal on Numerical Analysis, 61(5), 2023, pp. 2157-2181 |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 25 pages, 16 figures, 2 tables |
| Databáze: | arXiv |
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