This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approach-and in particular, FSMSN-results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance-i.e., measured by wall-clock time-will be studied in a subsequent publication.

翻译：本文致力于开发一种两步场分裂非线性预条件子，以加速非均匀多孔介质中两相流动与输运过程的收敛。我们提出一种名为"场分裂乘性施瓦茨牛顿法"（FSMSN）的场分裂算法，包含两步：首先，通过依次近似求解两个子问题，对压力和饱和度进行非线性预条件更新；随后，基于预条件状态下的系统线性化获得的牛顿更新，执行全局求解步骤。通过具有挑战性的测试案例，将FSMSN与现有场分裂预条件子——乘性施瓦茨预条件非精确牛顿法（MSPIN）以及标准求解策略（如顺序全隐式方法和全隐式方法）进行了对比。比较结果凸显了迎风格式对预条件子算法性能的影响，以及预条件步骤中子问题容差动态自适应的重要性。实验表明，两步非线性预条件方法（特别是FSMSN）相比SFI与FIM方法能够实现更快的全局迭代收敛。预条件子对计算性能（即壁钟时间）的影响将在后续论文中研究。