A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

Abstract

A novel free–energy stable discontinuous Galerkin method is developed for the Cahn–Hilliard equation with non–conforming elements. This work focuses on dynamic polynomial adaptation (p–refinement) and constitutes an extension of the method developed by Manzanero et al. (2020) [8], which makes use of the summation–by–parts simultaneous–approximation term technique along with Gauss–Lobatto points and the Bassi–Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates non–conforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free–energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaptation is introduced based on the knowledge of the location of the interface between phases. The adaptation methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We test the scheme for freestream preservation and primary quantity conservation on non–conforming curvilinear meshes. We solve a steady one–dimensional interface test case to initially examine the accuracy of the adaptation. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free–energy are recovered, with a 35% reduction of the degrees of freedom for the two–dimensional case considered and a 48% reduction for the three–dimensional case.