GPU Optimization of ILU‐Preconditioned GCR for Solving 19‐Diagonal Linear Equations in GRAPES

This article investigates the GPU optimization of solving 19‐diagonal asymmetric linear systems within the numerical weather prediction model GRAPES. Such systems are commonly encountered when solving partial differential equations on 3D structured grids using finite difference methods. The five‐diagonal patch‐ILU preconditioner, which retains the essential connection coefficient, is well‐suited for GPU platforms as it accelerates linear iterative convergence by approximately tenfold and offers a degree of parallelism. However, the forward‐backward substitution process, used to solve the upper and lower triangular equations generated by the five‐diagonal patch‐ILU preconditioner, remains a major performance bottleneck on the GPU due to serial data dependencies. We designed the Shuffle Thomas algorithm, leveraging the GPU's shuffle functionality for data reuse, achieving efficient memory coalescing and data reuse, significantly enhancing memory throughput. Further exploiting the diagonal direction's parallelism in the substitution process, we designed the Divided Shuffle Thomas algorithm, doubling the instruction‐level parallelism. This approach achieved a to speedup compared to cuSPARSE‐gpsv. Our GCR solver on the Hygon DCU platform demonstrated a 5.41 to 8.47 times performance improvement over the CPU implementation with the same number of computing nodes, achieving higher computational efficiency with fewer processes. This has the potential to significantly enhance the computational efficiency for high‐resolution numerical weather forecasting.