Compatible Energy Dissipation of the Variable-Step \({\boldsymbol{L1}}\) Scheme for the Space-Time Fractional Cahn-Hilliard Equation

Abstract

.We construct and analyze the variable-step \(L1\) scheme to efficiently solve the space-time fractional Cahn–Hilliard equation in two dimensions. The associated variational energy dissipation law in continuous form is developed for the space-time fractional model in continuous form, which is compatible with that of the classical Cahn–Hilliard model. Detailed energy stability and convergence analysis of the proposed fully discrete scheme are rigorously established. We further show that the discrete energy dissipation law of the space-time fractional Cahn–Hilliard model is asymptotically compatible with that of the backward Euler scheme for the classical Cahn–Hilliard model when the order of the time fractional derivative and that of the space fractional Laplacian both approach \(1^-\) . The proposed scheme is shown to achieve the optimal temporal convergence rate with the variable time steps numerically. More importantly, numerical results reveal that the energy dissipation rate depends on both the order of the time fractional derivative and that of the space fractional one. Specifically, the exponent of the power law for the energy dissipation scales linearly with the order of the space fractional Laplacian.Keywordsspace-time fractional Cahn–Hilliard equationvariable-step \(L1\) schemevariational energyconvergenceadaptive time-stepping strategyMSC codes35R1165M5065M12

Topics & Concepts