A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations (Communications in Computational Physics, 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 thosefor electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve theinternal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However, such schemes are typicallynot conservative for the total energy. Recently, significant progress has been madein developing cell-centered Lagrangian schemes which have several good propertiessuch as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the formof the total energy, therefore they cannot be directly applied to the solution of eitherthe single internal energy equation or the multiple internal energy equations withoutsignificant modifications. Such modifications, if not designed carefully, may lead tothe loss of some of the nice properties of the original schemes such as conservation ofthe total energy. In this paper, we establish an equivalency relationship between thecell-centered discretizations of the Euler equations in the forms of the total energy andof the internal energy. By a carefully designed modification in the implementation,the cell-centered Lagrangian scheme can be used to solve the compressible fluid flowwith one or multiple internal energy equations and meanwhile it does not lose its totalenergy conservation property. An advantage of this approach is that it can be easilyapplied to many existing large application codes which are based on the frameworkof 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 interms of symmetry preserving, accuracy and non-oscillatory performance.