In this work, we analyse space-time reduced basis methods for the efficient numerical simulation of hemodynamics in arteries. The classical formulation of the reduced basis (RB) method features dimensionality reduction in space, while finite differences schemes are employed for the time integration of the resulting ordinary differential equation (ODE). Space-time reduced basis (ST-RB) methods extend the dimensionality reduction paradigm to the temporal dimension, projecting the full-order problem onto a low-dimensional spatio-temporal subspace. Our goal is to investigate the application of ST-RB methods to the unsteady incompressible Stokes equations, with a particular focus on stability. High-fidelity simulations are performed using the Finite Element (FE) method and BDF2 as time marching scheme. We consider two different ST-RB methods. In the first one - called ST-GRB - space-time model order reduction is achieved by means of a Galerkin projection; a spatio-temporal velocity basis enrichment procedure is introduced to guarantee stability. The second method - called ST-PGRB - is characterized by a Petrov--Galerkin projection, stemming from a suitable minimization of the FOM residual, that allows to automatically attain stability. The classical RB method - denoted as SRB-TFO - serves as a baseline for the theoretical development. Numerical tests have been conducted on an idealized symmetric bifurcation geometry and on the patient-specific one of a femoropopliteal bypass. The results show that both ST-RB methods provide accurate approximations of the high-fidelity solutions, while considerably reducing the computational cost. In particular, the ST-PGRB method exhibits the best performance, as it features a better computational efficiency while retaining accuracies in accordance with theoretical expectations.

翻译：本文分析用于动脉血流动力学高效数值模拟的时空降基方法。传统降基（RB）方法采用空间维度的降阶处理，同时利用有限差分格式对所得常微分方程（ODE）进行时间积分。时空降基（ST-RB）方法将降阶范式拓展至时间维度，将全阶问题投影至低维时空子空间。本文旨在研究ST-RB方法在非定常不可压Stokes方程中的应用，特别关注其稳定性。高保真模拟采用有限元（FE）方法与BDF2时间推进格式执行。我们考虑两种不同的ST-RB方法：第一种称为ST-GRB，通过伽辽金投影实现时空模型降阶，并引入时空速度基函数增广过程以保障稳定性；第二种方法称为ST-PGRB，其特点是基于佩特罗夫-伽辽金投影，通过适当最小化全阶模型残差自动获取稳定性。经典RB方法（记为SRB-TFO）作为理论发展的基准。数值测试在理想对称分叉几何与患者特异性股腘动脉旁路移植体两种模型上进行。结果表明，两种ST-RB方法均能提供高保真解的精确近似，同时显著降低计算成本。其中ST-PGRB方法表现最优，其在保持符合理论预期的精度的同时，实现了更优的计算效率。