Parallel Shared-Memory Isogeometric Residual Minimization (iGRM) for Three-Dimensional Advection-Diffusion Problems

Łoś, Marcin; Muñoz‐Matute, Judit; Podsiadło, Krzysztof; Paszyński, Maciej; Pingali, Keshav

doi:10.1007/978-3-030-50436-6_10

Cited by 1 publication

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“…Recently, in the conference proceedings [30], we extended the explicit dynamics method [20,[26][27][28][29] into the implicit dynamics solver. The solver was applied for the advection-diffusion problem.…”

Section: Introductionmentioning

confidence: 99%

Bulletin of the Polish Academy of Sciences: Technical Sciences

Łoś¹,

Woźniak²,

Muga

et al. 2021

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ments, and the second one models the spontaneous propagation of pathogen particles in the air. Unfortunately, the three-dimensional time-dependent advection-diffusion equations are complicated to simulate. This is related to the high computational cost of the three-dimensional problem as well as the instabilities of the numerical methods. As the remedy to the first problem, we propose the alternating directions implicit solver. We also focus on the mathematical analysis of the simulation that presents the necessary conditions for its stability.The alternating directions implicit solver (ADI) was originally proposed for performing finite-difference simulations of time-dependent problems on regular grids. The first papers concerning the ADI method were published in 1960 [2-5]. The ADI with finite difference method is still popular for fast solutions of different classes of problems with finite difference method [6,7]. The method introduces intermediate time steps in its basic version, and the differential operator is split into sub-operators, containing only the x, y, z derivatives. The time integration scheme involves sub-steps with only one sub-operator on the left-hand side and the other sub-operators on the right-hand side, acting on the previous sub-step solutions. As a result of this direction splitting, after the discretization of the linear equations system, we deal only with derivatives in one direction while the rest of the operator is on the righthand side. If derived on the regular three-dimensional grid, the

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