Advances in the Free-Lagrange Method Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method pp 57-65 | Cite as

# An explicit-implicit solution of the hydrodynamic and radiation equations

## Abstract

A solution of the coupled radiation-hydrodynamic equations on a median mesh is presented for a transient, three-dimensional, compressible, multimaterial, free-Lagrangian code. The code uses fixed-mass particles surrounded by median Lagrangian cells. These cells are free to change connectivity, which ensures accuracy in the differencing of equations and allows the code to handle extreme distortions. All calculations are done on a median Lagrangian mesh that is constructed from the Delaunay tetrahedral mesh using the Voronoi connection algorithm. Because each tetrahedron volume is shared equally by the four mass points (computational cells) located at the tetrahedron vertices, calculations are done at a tetrahedron level for enhanced computational efficiency, and the rate-of-change data are subsequently accumulated at mass points from these tetrahedral contributions. The hydrodynamic part of the calculations is done using an explicit time-advancement technique, and the radiation calculations are done using a hybrid explicit-implicit time-advancement scheme in the equilibrium-diffusion limit. An explicit solution of the radiation-diffusion equation is obtained for cells that meet the current time-step criterion imposed by the hydrodynamic solution, and a fully implicit point-relaxation solution is obtained elsewhere without defining an inversion matrix. The approach has a distinct advantage over the conventional matrix-inversion approaches, because defining such a matrix for an unstructured grid is both cumbersome and computationally intensive. The new algorithm runs >20 times faster than a matrix-solver approach using the conjugate-gradient technique, and is easily parallelizable on the Cray family of supercomputers. With the new algorithm, the radiation-diffusion part of the calculation runs about twice as fast as the hydrodynamic part of the calculation. The code conserves mass, momentum, and energy exactly, except in some pathological situations.

## 1. Nomenclature

- a
_{R} Rosseland mean transport coefficient

- A
area vector

*c*_{0}light speed in vacuum

- D/Dt
substantial derivative,

*∂/∂t*+ q · ∇- e
specific total energy

- I
specific internal energy

- I
unit tensor

- k
thermal conductivity

- M
mass-point mass

- n
number of tetrahedra associated with a mass point

- p
pressure

- q
velocity vector

- q
^{111} heat-generation rate per unit volume

- S
material-stress tensor

- t
time

- T
temperature

- u,v,w
coordinate velocities

- V
volume

- x,y,z
coordinate directions

- δt
time-step size

- μ
fluid viscosity, including artificial viscosity

- ϱ
density

- σ
Stefan-Boltzmann constant

- σ
overall stress tensor, including pressure, viscous stress tensor, artificial-viscosity tensor, and material-stress tensor;

- τ
viscous and artificial-viscosity stress tensor

- ϕ
temperature to the fourth power,

*T*^{4}- b
median-mesh boundary

- f
free surface

- i,j,k,l
mass-point designations

- r
radiation

- t
tetrahedron

- x, y, z
coordinate directions

- l
iteration number

- n
old-time level

*n*+1new-time level

- *
either old- or new-time level

- —
mass-point average

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## References

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