# On the Class of Compact Grid-Characteristic Schemes

## Abstract

This work is devoted to the numerical solution of the linear hyperbolic transport equation. The importance of this topic is explained by the wide variety of possible applications: seismic survey of oil and gas deposits, seismic resistance of ground facilities estimation, hydrodynamic simulation. In one-dimensional case, a class of two-point compact difference schemes is considered. With the usage of appropriate procedure for polynomial interpolation and analysis of obtained solution behavior second-third order of approximation is achieved for continuous case. Proposed schemes also produce fewer oscillations on discontinuous solutions. This approach was generalized to the two-dimensional case. The technique of splitting in coordinates and parallelepiped meshes were successfully used. The influence of separate steps of the computational algorithm on the accuracy of the resulting scheme is investigated. The best results were obtained by storing orthogonal (additional) derivatives; however, it increased the random access memory consumption. With the same basic principle compact grid-characteristic schemes for the three-dimensional case were constructed. Due to the increase of the computational complexity of the algorithm the parallelization with Massively Parallel Processing (MPI) technology was done. To keep the order of approximation two more orthogonal derivatives must be stored. A set of computational simulation was carried out in this work for comparison with analytical solutions.

## Keywords

Computer simulation Compact schemes Grid-Characteristic method Transport equation## Notes

### Acknowledgements

This work was supported by the Russian Foundation for Basic Research, project no. 16-08-00212_a.

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