An Energy Stable Local Discontinuous Galerkin Method for the Isothermal Navier-Stokes-Korteweg Equations
Document Type
Article
Publication Date
8-2025
Department
Department of Mathematical Sciences
Abstract
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.
Publication Title
Journal of Scientific Computing
Recommended Citation
Tian, L.,
Xu, Y.,
Yang, Y.,
&
der Vegt, J.
(2025).
An Energy Stable Local Discontinuous Galerkin Method for the Isothermal Navier-Stokes-Korteweg Equations.
Journal of Scientific Computing,
104(2).
http://doi.org/10.1007/s10915-025-02979-x
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p2/1855