## Organization:

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| School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, Province, P.R. China. |

## Description:

In this research we will concentrate on designing an efficient high order accurate numerical methods for the non-isothermal Navier-Stokes-Korteweg (NSK) equations, which are an example of a diffuse interface model for multifluid transition problems and have widely been used to model the dynamics of a compressible fluid exhibiting phase transitions between liquid and vapor.

To achieve this, a local discontinuous Galerkin (LDG) method with a reduced number of auxiliary variables will be developed for the NSK equations in order to reduce the memory overhead of standard LDG discretizations. In addition, as the NSK equations contain a third order capillary term, which imposes a severe time step constraint for explicit time integration methods, it will be useful to consider implicit-explicit (IMEX) Runge-Kutta time integration methods that treat the convective term explicitly and the viscous and capillary terms implicitly. We will also try to find a stable discretization that can capture the thin interface using as few elements as possible. What’s more, the hpGEM discontinuous Galerkin toolkit will be used for the rapid parallel implementation of the DG discretization of the NSK equations in a new C++ code.