## Finite-Difference Methods For Solving Multidimentional Time-Dependent Transport Problems Implemented In SATURN Package

#### Rashit ShagalievVNIIEF

R. M. Shagaliev, A. V. Alekseyev, A. V. Gichuk, A. A. Nuzhdin, N. P. Pleteneva, L. P. Fedotova

Russian Federal Nuclear Center - All-Russian Scientific Research Institute of Experimental Physics (RFNC-VNIIEF), 607190, Mira av., 37, Sarov, Russia

The paper briefly describes the numerical methods for time-dependent problems implemented in SATURN code complex. This system of codes is intended for solving 2D and 3D both linear and nonlinear spectral transport problems encountered in physics of high densities and energies in multiple-group approximation.

The main principles forming the basis for the numerical methods of solving 2D time-dependent transport problems used in SATURN are, as follows:

1. Two types of spatial grids are used in SATURN to approximate the transport equation in space variables, they are:

- regular non-orthogonal grids of convex quadrangles;

- irregular non-orthogonal grids of arbitrary-shape convex polygons.

Several conservative finite-difference schemes have been proposed for the 2D transport equation approximation using the grids above, namely: schemes of the DSn-method type, scheme with introduction of closing relations basing on moment equations, adaptive methods with refined grids in phase space, etc. All of these schemes have a common feature consisting in that with the use of non-orthogonal grids they preserve some important features of DSn-schemes, such as the transport equation approximation within a single cell of phase space and, consequently, a possibility to resolve systems of grid equations using point-to-point computations. At the same time, they slightly differ from each other in the accuracy of approximating on essentially non-orthogonal grids, in monotony of grid solutions, and in some other features. The paper gives a more detailed formulation of the schemes mentioned above and presents some results of their computational investigations.

2. KM-method of accelerating convergence of simple iterations and some modifications of this method are successfully used in SATURN complex for a higher efficiency of solving systems of multiple-group grid transport equations in optically dense nonlinear problems. The paper gives brief description of KM-method and some examples of its estimated efficiency.

3. Finding solutions to many multidimensional time-dependent transport problems results in large computational burden. To solve such problems on modern multiprocessor platforms, a combined parallelization algorithm is implemented in SATURN that includes a capability of spatial small-block decomposition of a problem using non-orthogonal grids in combination with decomposition in energy and angular variables. The paper describes specific features of the algorithm. Issues of generalizing the methods and algorithms described to a 3D case are considered.

Examples of computations by SATURN in combination with hydrodynamic code complexes for 2D non-linear application problems of laser thermonuclear fusion are presented.

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