TY - JOUR
T1 - A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations
AU - Juan Cheng, Chi-Wang Shu & Qinghong Zeng
JO - Communications in Computational Physics
VL - 5
SP - 1307
EP - 1328
PY - 2012
DA - 2012/12
SN - 12
DO - http://doi.org/10.4208/cicp.150311.090112a
UR - https://global-sci.org/intro/article_detail/cicp/7336.html
KW - 
AB - <p style="text-align: justify;">Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion
(ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those
for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the
internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However, such schemes are typically
not conservative for the total energy. Recently, significant progress has been made
in developing cell-centered Lagrangian schemes which have several good properties
such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form
of the total energy, therefore they cannot be directly applied to the solution of either
the single internal energy equation or the multiple internal energy equations without
significant modifications. Such modifications, if not designed carefully, may lead to
the loss of some of the nice properties of the original schemes such as conservation of
the total energy. In this paper, we establish an equivalency relationship between the
cell-centered discretizations of the Euler equations in the forms of the total energy and
of the internal energy. By a carefully designed modification in the implementation,
the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow
with one or multiple internal energy equations and meanwhile it does not lose its total
energy conservation property. An advantage of this approach is that it can be easily
applied to many existing large application codes which are based on the framework
of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in
terms of symmetry preserving, accuracy and non-oscillatory performance.</p>