Short NoteExtension of Kershaw diffusion scheme to hexahedral meshes
Introduction
Numerical solution of the time dependent diffusion equation on non-orthogonal meshes in two spatial dimensions is an essential feature of typical Lagrangian radiation hydrodynamics simulation codes used for designing inertial confinement fusion targets and modeling other high energy density plasmas. Electron thermal conduction is treated with a non-linear diffusion equation and radiative transfer is often modeled using multi-group flux-limited diffusion. For two dimensions, an initially orthogonal mesh can become non-orthogonal due to hydrodynamic flow irregularities and the diffusion equation must be numerically differenced on this non-orthogonal mesh. A novel approach to this problem was reported by Kershaw [1], where he used a variational method to derive the difference operator corresponding to the continuous diffusion operator on a non-orthogonal r–z mesh. Kershaw’s method leads to a nine-point differencing stencil and tractable positive definite matrix solutions for problems of practical significance. It reduces to the standard five-point differencing stencil for orthogonal meshes. For this reason Kershaw’s method is often used as a benchmark for comparison of more recent, higher order methods [2].
In this paper, the discretization scheme developed by Kershaw is extended to three dimensional non-uniform hexahedral x–y–z meshes. As three dimensional radiation hydrodynamics simulations become more commonplace, this three dimensional Kershaw scheme can be a viable approach to solving the diffusion equation. While higher order discretizations exist, the 3D Kershaw method has the benefit of using only zone centered unknowns with a local computational stencil making it relatively easy to implement in Lagrangian codes. Along these lines, the detailed difference equations for the resulting 19 point computational stencil are presented in this paper. While this scheme has limited accuracy, it can still serve as a benchmark for comparison of more elaborate higher order, but more expensive schemes.
This extension shares many of the same properties as the original method. The resulting matrix has the benefit of being symmetric positive definite (SPD). However, it is not an “M-matrix” and therefore negative answers are not mathematically forbidden. The convergence properties are also similar to the original method. In general, first order convergence is expected, except on non-smooth meshes where the accuracy is degraded. A numerical test is presented which demonstrates this property. Furthermore, the difference equations reduce to the standard second-order, 7-point scheme as the mesh becomes orthogonal and uniform, therefore higher order accuracy is expected in this case.
Section snippets
Discretizing the diffusion operator
The process for discretizing the diffusion operator closely follows Kershaw’s derivation of the two dimensional r–z method. The first step involves expressing the diffusion operator as a function of continuous variables K, L, and M as shown in (1) through (3). The variable, j, defined in (3) is the Jacobian which transforms from the physical variables (X, Y, Z) to (K, L, M), and is simply the zone volume. The lack of the factors of R that appear in Kershaw’s original derivation is the key
Computational results
The matrix B used to compute the diffusion matrix is defined asand as in the 2D method [1], by defining B as a function of the vectors RK, RL, and RM an assumption is made that the coordinates are a smooth function of the logical variables K, L, and M. This leads to a requirement that the mesh lines be smooth in order to obtain first order accuracy, which was demonstrated in [2] for the 2D method. In this section, the result of a convergence test is shown on
Conclusions
Expressions for the matrix elements of a symmetric positive definite matrix A that approximates the diffusion operator on non-orthogonal 3D x–y–z meshes are derived using the Kershaw formulation. This results in a 19-point stencil. Convergence testing of this scheme on non-orthogonal 3D meshes confirms that its properties are consistent with those of the original Kershaw scheme. The method is relatively simple to implement in simulation codes, which comes at the cost of accuracy. Specifically,
Acknowledgments
This work was supported by the University of Rochester Laboratory for Laser Energetics. The authors wish to thank Marvin Adams and James Morel for their help and advice.
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