We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

我们探索并开发了一种基于适当正交分解 (POD) 的放气方法，用于解决病态线性系统，出现在通过高度异质多孔介质的两相流模拟中。我们加速了预条件共轭梯度 (PCG) 方法的收敛，实现了高达五个因子的加速。所提出方法的前期额外计算成本取决于紧缩向量的数量。基于POD的紧缩方法针对特定问题和线性求解器进行了测试；然而，它可以应用于各种瞬态问题，并与多种求解器相结合，例如 Krylov 子空间和多重网格方法。