An Energy Stable Local Discontinuous Galerkin Method for the Isothermal Navier-Stokes-Korteweg Equations

In this article, we develop an energy stable local discontinuous Galerkin (LDG) method for the isothermal Navier-Stokes-Korteweg (NSK) equations. Since the test and trial functions in LDG discretisations have to be in the same finite element space, it is difficult to obtain energy stable LDG discretizations for the isothermal NSK equations. To bridge this gap we first write the pressure into a free energy function form and introduce the velocity as a variable, then we use an extra auxiliary variable containing both the free energy function and the square of the velocity. These auxiliary variables are chosen in the stability analysis as test functions for the density and momentum balance equations. Using the Crank-Nicolson (CN) time integration method, we can prove then the stability of the CN-LDG method. Numerical experiments are provided to demonstrate the theoretical results, in particular on adaptive meshes.