本研究發展一套基於 GPU 加速之 Building Cube Method（BCM）計算流體力學（Computational Fluid Dynamics, CFD）求解器，並以 Taylor-Green vortex 與圓柱繞流作為驗證案例，評估數值方法之正確性、能量耗散特性與平行計算效能。空間離散採用 Roe flux 計算數值通量，時間積分則分別實作顯式 Runge-Kutta 方法與隱式 LU-SGS（Lower-Upper Symmetric Gauss-Seidel）方法。
在顯式方法方面，本研究以 16×16×16 cells 作為基本 cube 進行 GPU 平行化，並透過與文獻之動能衰減曲線比較，驗證程式於 Taylor-Green vortex 問題中的數值準確性與能量耗散行為。針對隱式 LU-SGS 方法，由於前後向掃描具有資料相依特性，限制 GPU 平行效率，因此進一步將計算域拆分為 4×4×4 之 small-block，以提升計算粒度與平行並行度。
為降低 cube 細分後所增加之 halo communication 開銷，本研究設計專用資料陣列儲存所有 cube 資訊，使各 cube 在 halo region 計算時可直接存取陣列資料，而無需頻繁進行 GPU 或 MPI 通訊。此方法雖增加 GPU 記憶體使用量，但可有效降低通訊成本，提升 GPU 資源利用率與整體平行效率，使隱式 LU-SGS 求解器GPU、多節點環境下仍具良好擴充性。
平行架構採用 MPI 進行跨節點運算，每節點配置 8 張 GPU，目前已擴展至 32 GPU。單一 GPU 約負責 71,991,296個網格，整體模擬規模可達 2,303,721,472 個網格。研究進一步進行 strong scaling 與 weak scaling 測試，並透過時間分解分析量化計算與通訊比例。結果顯示，本研究之 GPU BCM CFD 平台於 weak scaling 下可維持良好平行效率，而 strong scaling 效率下降則主要來自 halo exchange 通訊負擔增加，且於多階段顯式 Runge-Kutta 方法中特別明顯。整體而言，small-block LUSGS 策略可有效提升隱式求解器之平行效率與擴充能力，證明本研究所發展之 GPU BCM CFD 平台具備大規模高解析流場模擬之能力與可擴展性。

This study develops a GPU-accelerated Building Cube Method (BCM) based Computational Fluid Dynamics solver and validates its numerical accuracy and parallel performance using Taylor–Green vortex and circular cylinder flow cases. The numerical scheme employs Roe flux for spatial discretization, while both explicit Runge–Kutta method and implicit LU-SGS methods are implemented for time integration.

For the explicit solver, GPU parallelization is performed using 16×16×16-cell cubes, and numerical accuracy is verified through kinetic energy decay comparisons with reference data. For the implicit LU-SGS solver, a 4×4×4 small-block strategy is introduced to im-prove parallelism and computational efficiency by reducing data dependency during for-ward and backward sweeps.

To minimize halo communication overhead, a dedicated data array is designed to store cube information, allowing direct data access during halo-region computations and reducing communication costs. This approach improves GPU resource utilization and overall parallel efficiency.

The solver is implemented on a multi-GPU cluster using MPI, scaling up to 32 GPUs with a total problem size of 2,303,721,472 cells. Strong and weak scaling tests show good scalability, particularly for weak scaling. The proposed small-block LU-SGS strategy significantly enhances the scalability of the implicit solver, demonstrating the capability of the developed GPU BCM platform for large-scale, high-resolution flow simulations.

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