The energy dissipation is an important and essential property of classical phase-field equations . However, it is still unknown if the phase-field models with Caputo time-fractional derivative preserve this property, which is challenging due to the existence of both nonlocality and nonlinearity. Our recent work shows that on the continuous level, the time-fractional energy dissipation law and the weighted energy dissipation law can be achieved. Inspired by them, we study in this article the energy dissipation of some numerical schemes for time-fractional phase-field models , including the convex-splitting scheme, the stabilization scheme, and the scalar auxiliary variable scheme. Based on a lemma about a special Cholesky decomposition, it can be proved that the discrete fractional derivative of energy is nonpositive, i.e., the discrete time-fractional energy dissipation law , and that a discrete weighted energy can be constructed to be dissipative, i. e., the discrete weighted energy dissipation law. In addition, some numerical tests are provided } to verify our theoretical analysis.
The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy law. The extension for a (2- \alpha})-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried to verify our theoretical analysis.
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energy dissipation is an important and essential property of classical
numerical integration of
R
. However, it is still unknown if the phase-field models with Caputo time-fractional derivative preserve this property, which is challenging due to
is a delicate task which needs to recover at the discrete level intrinsic properties of
R
existence of both nonlocality and nonlinearity. Our recent work shows that on the continuous level, the time-fractional energy dissipation law and
solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established,
R
weighted energy dissipation law can be achieved. Inspired by them, we study in this article the energy dissipation of some numerical schemes
corresponding theory
R
, including the convex-splitting scheme, the stabilization scheme, and the scalar auxiliary variable scheme. Based on a lemma about a special Cholesky decomposition, it can be proved that the discrete fractional derivative of energy is nonpositive, i.e., the discrete time-fractional energy dissipation law , and that
is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and
R
can be constructed to be dissipative, i. e., the discrete weighted energy dissipation law. In addition, some numerical tests are provided
law. The extension for a (2-
A
\alpha
A
)-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried