Commun. Comput. Phys., 12 (2012), pp. 1307-1328.

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Juan Cheng 1*, Chi-Wang Shu 2, Qinghong Zeng 3

1 National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China.
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
3 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China.

Received 15 March 2011; Accepted (in revised version) 9 January 2012
Available online 8 May 2012


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.

AMS subject classifications: 65M06, 76M20

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Key words: Lagrangian scheme, conservative, cell-centered, internal energy equation, compressible fluid flow, three-temperature model.

*Corresponding author.
Email: (J. Cheng), (C.-W. Shu), (Q. Zeng)

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