## Efficient solution techniques for two-phase flow in heterogeneous porous media using exact Jacobians.(English)Zbl 1453.65063

Summary: 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 $$\mathrm{CO_2}$$ 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.

### MSC:

 65F08 Preconditioners for iterative methods 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 76S05 Flows in porous media; filtration; seepage 76T10 Liquid-gas two-phase flows, bubbly flows

### Software:

BoomerAMG; CPR-AMG; hypre; IPARS; TAPENADE; PETSc
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