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Algebraic multigrid for computational fluid dynamics simulations: An overview of cycle choice and performance analysis
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DOI：10.1016/j.compfluid.2026.107205.png)
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Abstract
En 中文
The paper investigates the performance of different multigrid cycle types for the solution of a linearised implicitly coupled pressure–velocity system for incompressible flow, in the context of the classical algebraic multigrid method. Since the block-coupled linear system comprises the parabolic equations for three velocity components and the elliptic Poisson equation for pressure, the multigrid setup phase is conducted using a constructed primary matrix corresponding to the pressure equation without the velocity cross-coupling terms. Thus, the selection of representative equations and calculation of coarse multigrid levels relies only on the discretised pressure equation, and it is assumed that this choice of the primary matrix, and consequently the interpolation and restriction weights in the multigrid cycle, will not negatively affect the convergence of the velocity solution components. The full multigrid cycle (FMG) was implemented, and its performance was compared to the standard V-, W and F-cycles for various test cases: two-dimensional backward facing step; and cases from real industrial applications: BB2 submarine and the DrivAer model car. The results show that cycle performance depends strongly on mesh topology and the resulting coefficient matrix structure. For block-structured hexahedral meshes, all cycle types converge; the FMG cycle achieves the lowest computational cost per linear iteration, while the V-cycle provides the best total time-to-solution due to fewer non-linear iterations required. For unstructured tetrahedral meshes, only the V-cycle converges throughout the simulation, while the W-cycle diverges at an intermediate non-linear iteration and the F- and FMG cycles diverge at the first non-linear iteration. The divergence is attributed to rapid bandwidth growth in coarse-level matrices produced by the SAMG coarsening procedure on tetrahedral meshes, which introduces errors into the incomplete LU smoother. A per-level spectral diagnostic is introduced, tracking the dominant eigenvalue of the coarse-level operator at each level of the AMG hierarchy across non-linear iterations. The diagnostic measures the spectral properties of coarse-level operators across the three test cases. The observed eigenvalue growth correlates with the cycle-type-dependent convergence behaviour, and the results suggest that mesh quality metrics such as aspect ratio and non-orthogonality are related to AMG coarse-level conditioning. A domain decomposition study on the BB2 submarine case shows that graph-based decomposers achieve lower AMG iteration counts than geometric decomposition strategies under matched load balance.
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