Two efficient and scalable numerical solution methods will be compared using exact Jacobians to solve the fully coupled Newton systems arising during fully implicit discretization of the equations for two-phase flow in porous media. These methods use algebraic multigrid (AMG) to solve the linear systems in every Newton step. The algebraic multigrid methods rely on (i) a Schur Complement Reduction (SCR-AMG) and (ii) a Constrained Pressure Residual method (CPR-AMG) to decouple elliptic and hyperbolic contributions. Both methods employ automatic differentiation (AD) to calculate exact Jacobians and are compared with finite difference (FD) approximations of the Jacobian. The superiority of AD is shown by several numerical test cases from the field of CO₂ geo-sequestration comprising two- and three-dimensional examples. A weak scaling test on JUQUEEN, a BlueGene/Q supercomputer, demonstrates the efficiency and scalability of both methods. To achieve maximum comparability and reproducibility, the Portable Extensible Toolkit for Scientific Computation (PETSc) is used as framework for the comparison of all solvers.

将使用精确的雅可比矩阵对两种有效且可扩展的数值求解方法进行比较，以求解在多孔介质中两相流方程的完全隐式离散化过程中产生的完全耦合的牛顿系统。这些方法使用代数多重网格（AMG）来求解牛顿每一步中的线性系统。代数多重网格方法依赖于（i）减少Schur补数（SCR-AMG）和（ii）约束压力残差法（CPR-AMG）来解耦椭圆和双曲贡献。两种方法都使用自动微分（AD）来计算精确的Jacobian值，并与Jacobian的有限差分（FD）近似值进行比较。AD的优越性通过CO ₂领域的几个数值测试案例得到了证明。地理隔离，包括二维和三维示例。在BlueGene / Q超级计算机JUQUEEN上进行的弱缩放测试证明了这两种方法的效率和可伸缩性。为了获得最大的可比性和可重复性，便携式可扩展科学计算工具包（PETSc）被用作比较所有求解器的框架。