2025/10/13 by Meng, Jiale, Tang, Shuqi, Wise, Steven M. +1
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2510.11152
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 81922 grid, the RTX~4090 achieves 61× over a single-core CPU, and in 3D at 5123, 46×. 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 5123 Navier-Stokes simulations on a single GPU. Grain growth on a 5122 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× 256× 1024 grid, agreeing with experimental observations.