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

Lagrangianmethods arewidely used inmany fields formulti-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 staggeredgrid 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.

Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations.To discretize this system of conservation laws,we exploit the(adjoint consistent)symmetric version of the interior penalty discontinuous Galerkin finite element method.To define the necessary coarse-level solver required for the definition of the proposed preconditioner,we exploit ideas from composite finite element methods,which allow for the definition of finite element schemes on general meshes consisting of polygonal(agglomerated)elements.The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two-and three-dimensions.

SPECTRAL-DIFFERENCE METHOD FOR TWO-DIMENSIONAL COMPRESSIBLE FLUID FLOW

We develop a combined Fourier spectral-finite difference method for solving 2-dimensional, semi-periodic compressible fluid how problem. The error estimation, as well as the convergence rate, is presented.