发表机构
PowerChina Kunming Engineering Corporation Limited; College of Water Conservancy and Hydropower Engineering, Hohai University（中国电建昆明工程有限公司; 河海大学水利水电学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
OptiXDE 是受傅里叶光学启发的无矩阵谱算子框架，适用于均匀网格和嵌入域微分方程，计算高效，在多类问题中表现优异，加速效果显著。
AI 中文摘要
OptiXDE 是一种用于均匀网格和嵌入域上微分方程的无矩阵谱算子框架。受傅里叶光学中角谱传播的启发，它将变换对角空间算子映射为解析模态乘数，并将其与物理空间算子组合以处理非线性项、几何结构和边界条件。研究人员在瞬态扩散、周期及嵌入域泊松问题、三次非线性薛定谔方程、粘性 Burgers 动力学、二维 Allen-Cahn 方程，以及从 Taylor-Green 涡到嵌入圆柱涡脱的不可压缩流动中，均验证了通用“变换-算子-逆变换”框架的适用性。对于与变换兼容的线性问题，其解的精度接近浮点极限；而在奇异 L 形域上，误差仅局部出现在凹角和正则化界面附近。非线性基准测试显示其达到二阶时间收敛性，并呈现预期的保守或耗散行为；在长时间涡脱过程中，不可压缩性始终保持在舍入误差水平。无矩阵更新的计算复杂度为 O(N log N)，内存复杂度为 O(N)。驻留设备的变换工作负载实现了 94.9 倍的 GPU 加速，完整的嵌入圆柱求解器在匹配数值设置下实现了 42.1 倍的 CPU-GPU 加速。这些结果表明，OptiXDE 是一种确定性、可扩展的以算子为中心的框架，适用于结构化和嵌入域上的微分方程。
英文摘要
OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform--operator--inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen--Cahn equation and incompressible flows from the Taylor--Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require \(\mathcal{O}(N\log N)\) work and \(\mathcal{O}(N)\) memory. Device-resident transform workloads reach \(94.9\times\) GPU acceleration, and the complete embedded-cylinder solver achieves a \(42.1\times\) CPU--GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.
Comments26 pages; 7 figures
