Geodesic
Solving grid equations using the alternating-triangular method on a graphics accelerator
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 12 (2023) no. 2, pp. 78-92

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The paper describes a parallel-pipeline implementation of solving grid equations using the modified alternating-triangular iterative method (MATM), obtained by numerically solving the equations of mathematical physics. The greatest computational costs at using this method are on the stages of solving a system of linear algebraic equations (SLAE) with lower triangular and upper non-triangular matrices. An algorithm for solving the SLAE with a lower triangular matrix on a graphics accelerator using NVIDIA CUDA technology is presented. To implement the parallel-pipeline method, a three-dimensional decomposition of the computational domain was used. It is divided into blocks along the $y$ coordinate, the number of which corresponds to the number of GPU streaming multiprocessors involved in the calculations. In turn, the blocks are divided into fragments according to two spatial coordinates — $x$ and $z$. The presented graph model describes the relationship between adjacent fragments of the computational grid and the pipeline calculation process. Based on the results of computational experiments, a regression model was obtained that describes the dependence of the time for calculation one MATM step on the GPU, the acceleration and efficiency for SLAE solution with a lower triangular matrix by the parallel-pipeline method on the GPU were calculated using the different number of streaming multiprocessors.
Keywords: mathematical modeling, parallel algorithm, graphics accelerator. 
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     author = {A. I. Sukhinov and V. N. Litvinov and A. E. Chistyakov and A. V. Nikitina and N. N. Gracheva and N. B. Rudenko},
     title = {Solving grid equations using the alternating-triangular method on a graphics accelerator},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a Vy\v{c}islitelʹna\^a matematika i informatika},
     pages = {78--92},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURV_2023_12_2_a3/}
}
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A. I. Sukhinov; V. N. Litvinov; A. E. Chistyakov; A. V. Nikitina; N. N. Gracheva; N. B. Rudenko. Solving grid equations using the alternating-triangular method on a graphics accelerator. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 12 (2023) no. 2, pp. 78-92. http://geodesic.mathdoc.fr/item/VYURV_2023_12_2_a3/