| Autor: |
REN-JUN QI, XUAN ZHAO |
| Předmět: |
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| Zdroj: |
SIAM Journal on Scientific Computing; 2024, Vol. 46 Issue 1, pA130-A155, 26p |
| Abstrakt: |
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 longtime 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. [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
| Externí odkaz: |
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