Uniform $L^p$-Bound of the Allen-Cahn Equation and Its Numerical Discretization

Abstract

We study uniform bounds associated with the Allen-Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, shares the uniform $L^p$-bound for all even $p$, which also leads to the maximum principle. In comparison, a couple of other spatial discretization schemes, namely the Fourier spectral Galerkin method and spectral collocation method preserve the $L^p$-bound only for $p = 2$. Moreover, fully discretized schemes based on the Fourier collocation method for spatial discretization and Strang splitting method for time discretization also preserve the uniform $L^2$-bound unconditionally.