Energy Conservative Local Discontinuous Galerkin Methods for the Euler-Korteweg Equations

Abstract

In this paper, we develop through a careful selection of the auxiliary variables and numerical fluxes an energy conservative local discontinuous Galerkin (LDG) method based on a hybrid form of the general Euler-Korteweg (EK) equations with a variable capillarity coefficient. This energy conservative LDG discretization is of optimal order of accuracy for alternating numerical fluxes, but not for central numerical fluxes which result in the reduction of one order of accuracy when odd degree polynomial basis functions are used. Also, a relatively simple energy conservative LDG discretization for the EK-equations with an irrotational velocity field is presented. Due to the presence of a highly nonlinear third-order spatial derivative term, which originates from the divergence of the Korteweg stress tensor, we employ the novel semi-implicit spectral deferred correction (SDC) method as temporal discretization. The SDC method can be applied to highly nonlinear ordinary differential equations (ODEs) without separating stiff and non-stiff components and is numerically stable for a time step proportional to the mesh size. Numerical experiments, including ones with adaptive meshes, are performed to illustrate the accuracy and capability of the proposed methods to solve the EK-equations.