A GPU-Accelerated Matrix-Free FAS Multigrid Solver for Navier-Stokes Equations with Memory-Efficient Implementations
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| Format: | Preprint |
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2025
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| author | Meng, Jiale Tang, Shuqi Wise, Steven M. Guo, Zhenlin |
| author_facet | Meng, Jiale Tang, Shuqi Wise, Steven M. Guo, Zhenlin |
| contents | We develop a matrix-free Full Approximation Storage (FAS) multigrid solver based on staggered finite differences and implemented on GPU in MATLAB. To enhance performance, intermediate variables are reused, and an X-shape Multi-Color Gauss-Seidel (X-MCGS) smoother is introduced, which eliminates conditional branching by partitioning the grid into four submatrices. Restriction and prolongation operators are also GPU-accelerated. Convergence tests verify robustness and accuracy, while benchmarks show substantial speedups: for the 2D heat equation on an $8192^2$ grid, the RTX~4090 achieves $61\times$ over a single-core CPU, and in 3D at $512^3$, $46\times$. A memory-efficient implementation of first- and second-order projection schemes reduces GPU-resident variables from 12/15 to 8, lowering memory footprint and improving performance by 20--30%, enabling $512^3$ Navier-Stokes simulations on a single GPU. Grain growth on a $512^2$ grid accommodates up to $q=1189$ (2D) and $q=123$ (3D) orientations, reproducing expected scaling laws. Coupled with Cahn-Hilliard equations, air-water two-bubble coalescence is simulated on a $256\times 256\times 1024$ grid, agreeing with experimental observations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A GPU-Accelerated Matrix-Free FAS Multigrid Solver for Navier-Stokes Equations with Memory-Efficient Implementations Meng, Jiale Tang, Shuqi Wise, Steven M. Guo, Zhenlin Numerical Analysis We develop a matrix-free Full Approximation Storage (FAS) multigrid solver based on staggered finite differences and implemented on GPU in MATLAB. To enhance performance, intermediate variables are reused, and an X-shape Multi-Color Gauss-Seidel (X-MCGS) smoother is introduced, which eliminates conditional branching by partitioning the grid into four submatrices. Restriction and prolongation operators are also GPU-accelerated. Convergence tests verify robustness and accuracy, while benchmarks show substantial speedups: for the 2D heat equation on an $8192^2$ grid, the RTX~4090 achieves $61\times$ over a single-core CPU, and in 3D at $512^3$, $46\times$. A memory-efficient implementation of first- and second-order projection schemes reduces GPU-resident variables from 12/15 to 8, lowering memory footprint and improving performance by 20--30%, enabling $512^3$ Navier-Stokes simulations on a single GPU. Grain growth on a $512^2$ grid accommodates up to $q=1189$ (2D) and $q=123$ (3D) orientations, reproducing expected scaling laws. Coupled with Cahn-Hilliard equations, air-water two-bubble coalescence is simulated on a $256\times 256\times 1024$ grid, agreeing with experimental observations. |
| title | A GPU-Accelerated Matrix-Free FAS Multigrid Solver for Navier-Stokes Equations with Memory-Efficient Implementations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.11152 |