Legendre spectral volume methods for Allen–Cahn equations by the direct discontinuous Galerkin formula

Abstract

In this paper, we introduce novel class of Legendre spectral volume (LSV) methods for solving Allen–Cahn equations. Each spectral volume (SV) is refined with $k$ Gauss–Legendre points to define an arbitrary order control volume (CV). Moreover, the second derivative is handled using the direct discontinuous Galerkin (DDG) approach. Furthermore, four numerical experiments are detailed including 1D and 2D Allen–Cahn equations with Neumann and periodic boundary conditions. These experiments demonstrate the stability and accuracy in capturing phase transitions of the approach. Meanwhile, we also show the LSV methods can maintain physical properties such as energy dissipation and uniform boundedness. It is worth mentioning that we observe that the LSV methods achieve both optimal convergence and superconvergence as the numerical flux parameter is carefully selected.