David S. Kershaw

The incomplete Cholesky—conjugate gradient method for the iterative solution of systems of linear equations☆

A new iterative method for the solution of systems of linear equations has been recently proposed by Meijerink and van der Vorst [1]. This method has been applied to real laser fusion problems taken from typical runs of the laser fusion simulation code LASNEX [2]. These same problems were also solved by various standard iteration methods. On a typical hard problem, the new method is about 8000 times faster than the point Gauss-Seidel method, 200 times faster than the alternating direction implicit method, and 30 times faster than the block successive overrelaxation method with optimum relaxation factor. The new method has two additional virtues. (1) Most of the algorithm is trivially vectorizable with a vector length equal to the full dimension of the system of linear equations. Thus, great savings are possible on vector machines. (2) The new method has a universal scope of application for solution of implicitly differenced partial differential equations. The only restrictions are that the matrix be symmetric and positive definite. The algorithm of Meijerink and van der Vorst which applied only to positive definite symmetric M-matrices is generalized to apply to positive definite symmetric matrices and further generalized to apply to nonsingular matrices arising from partial differential equations. A general description of the method is given. Numerical results are discussed and presented, and an explanation is given for the success of the method.

Differencing of the diffusion equation in Lagrangian hydrodynamic codes

Abstract The general problem of finite differencing the diffusion equation on a two-dimensional Lagrangian hydrodynamic mesh is discussed and a set of general criteria is developed. A detailed description is given of a particular difference scheme satisfying these criteria. A numerical test case is presented.

3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method

Abstract We describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using discontinuous finite element space and an explicit Runge-Kutta time discretization. This scheme combines the accuracy of a higher-order Godunov scheme with the unstructured mesh capabilities of finite elements that can be explicitly evolved in time. The spatial discretization uses trilinear isoparametric elements (tetrahedrons, pyramids, prisms and hexahedrons) in which the primitive variables (mass density, velocity and pressure) are piecewise trilinear. Upwinding is achieved by using Roes characteristic decomposition of the inter-element boundary flux depending on the sign of characteristic wave speeds. The characteristics are evaluated at the Roe average, of variables on both sides of the inter-element boundary, for a general equation of state. An explicit second order Runge-Kutta time stepping is used for the time discretization. To capture shocks, we have generalized van Leers 1D nonlinear minmod slope limiter to 3D using a quadratic programming scheme. For very strong shocks we find it necessary to supplement this with a Godunov stabilization where the trilinear representation of the variables is reduced to its constant average value. The resulting numerical scheme has been tested on a variety of problems relevant to ICF (inertial confinement fusion) target design and appears to be robust. It accurately captures shocks and contact discontinuities without unstable oscillations and has second-order accuracy in smooth regions. Object-oriented programming with the C++ programming language was used to implement our numerical scheme. The object-oriented design allows us to remove the complexities of an unstructured mesh from the basic physics modules and thereby enables efficient code development.

Solution of Single Tridiagonal Linear Systems and Vectorization of the ICCG Algorithm on the Cray-1

Publisher Summary This chapter presents the solution of single tridiagonal linear systems and vectorization of the ICCG algorithm on the Cray-1. When the numerical algorithms used to solve the physics equations in codes which model laser fusion are examined, it is found that a large number of subroutines require the solution of tridiagonal linear systems of equations. Radiation transport, thermal- and suprathermal-electron transport, ion thermal conduction, charged-particle, and neutron transport require the solution of tridiagonal systems of equations. The standard algorithm that has been used in the past on CDC 7600s will not vectorize and hence, cannot take advantage of the large speed increases possible on the Cray-1 through vectorization. There is an alternative algorithm for solving tridiagonal systems called cyclic reduction, which allows for vectorization, and is optimal for the Cray-1. Software based on this algorithm is being used in LASNEX to solve tridiagonal linear systems in the subroutines. The new algorithm runs five times faster than the standard algorithm on the Cray-1.

Solution of the diffusion equation by finite elements in hydrodynamic codes

Abstract The radiation diffusion equation is solved by the finite element method. The energy densities are point centered. These are integrated into a program architecture which requires zonally averaged quantities. Numerical results are presented which compare the scheme with existing finite difference techniques for zonal variables. Finite elements give better results for transport dominated problems on non-orthogonal meshes such as might be generated by Lagrangian hydrodynamic distortions.

A simple and fast method for computing the relativistic Compton scattering kernel for radiative transfer

Abstract The Klein-Nishina differential cross section averaged over a relativistic Maxwellian electron distribution is analytically reduced to a single integral, which can then be rapidly evaluated in a variety of ways. A particularly fast method for numerically computing this single integral is presented. This is, to our knowledge, the first correct computation of the Compton scattering kernel.

Evaluation of integrals of the Compton scattering cross-section

Abstract Exact formulae are derived for various integrals of the Compton scattering cross-section. The interaction kernel is integrated over outgoing photon frequency and direction, and over a relativistic Maxwellian distribution for the electrons. The total Compton cross-section, the energy exchange rate, and the transport mean free path are thereby expressed in terms of single integrals of analytic functions. In addition, these integrals produce simple analytic expressions in the limiting cases of either small or large frequency or electron temperature. A numerical method based on Gaussian quadrature is used to compute the transport mean free path. A comparison with previously published results is presented.

Diffusion coefficient for the Compton Fokker-Planck equation☆

Abstract An exact analytical formula for the diffusion coefficient of the Compton Fokker-Planck equation is derived. The formula is valid for arbitrary values of the electron temperature and photon energy. For applications in production-level radiation transport codes, a fast numerical method to compute the coefficient is presented.

A fast method for computing the integrals of the relativistic Compton scattering kernel for radiative transfer

Abstract For various computer simulation applications, one needs the integrals of the Compton scattering kernel over its parametes. An efficient and accurate for evaluaying these integrals is described and the corresponding software is available upon request.

3D unstructured mesh ALE hydrodynamics with the upwind discontinuous galerkin method

The authors describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using a discontinuous Galerkin method (DGM) and an explicit Runge-Kutta time discretization. Upwinding is achieved through Roes linearized Riemann solver with the Harten-Hyman entropy fix. For stabilization, a 3D quadratic programming generalization of van Leers 1D minmod slope limiter is used along with a Lapidus type artificial viscosity. This DGM scheme has been tested on a variety of hydrodynamic test problems and appears to be robust making it the basis for the integrated 3D inertial confinement fusion modeling code (ICF3D). For efficient code development, they use C++ object oriented programming to easily separate the complexities of an unstructured mesh from the basic physics modules. ICF3D is fully parallelized using domain decomposition and the MPI message passing library. It is fully portable. It runs on uniprocessor workstations and massively parallel platforms with distributed and shared memory.