High order conservative Lagrangian scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations are widely used in modeling various optically thick high-energy-density-physics environments, such as those in astrophysics and inertial confinement fusion (ICF). In this paper, we will discuss the methodology to construct a high order conservative Lagrangian scheme solving 3-T RH equations. Specifically, the three new energy variables are defined first, in the form of which the three energy equations of the 3-T RH equations are rewritten. The main advantage of this formulation is that it facilitates the design of a scheme with both conservative property and arbitrary high order accuracy. Starting from one dimension and based on the multi-resolution WENO reconstruction and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a third order conservative Lagrangian scheme both in space and time. To determine the numerical flux for the conservative advection terms in the 3-T RH equations, we propose a HLLC numerical flux which is derived from the divergence theorem rigorously and is suitable for multi-material problems with the ideal-gas equations of state. After that, we discuss how to design a class of high order positivity-preserving explicit Lagrangian schemes to solve the 3-T RH equations which only contain the conservative advection terms in space. Preliminary extension to the two dimensional case is also considered. Finally, various numerical tests are given to verify the desired properties of the high order Lagrangian schemes such as high order accuracy, non-oscillation, conservation and adaptation to multi-material problems.