Abstract
Multiscale methods for solving strongly heterogenous systems in reservoirs have a long history from the early ideas used on incompressible flow to the newly released version in commercial simulation. Much effort has been put into making the MsFV method work for fully unstructured multiphase problems. The MsRSB version is a newly developed version, which tackles most of the "real" world problems. It is to our knowledge, the only multiscale method that has been released in a commercial simulator. You can alternatively see the method as a variant of smoothed aggregation or as an iterative approach to AMG with energy minimizing basis functions. This will be discussed in detail.
So far, most work on comparing MsRSB with AMG methods has been on qualitative performance measures like iteration number rather than on pure runtime on fair code implementation. We discuss the theoretical performance and show the practical performance for our implementation. Here, we compare performance of pure AMG, standard two-level MsRSB with pure AMG as coarse solver, as well as a new truly multilevel MsRSB scheme. Our implementation uses the DUNE-ISTL framework. To limit the scope of the discussion we restrict our assessment to AMG with aggregation and smoothed aggregation and the MsRSB method. These three methods are closely related and are primarily distinguished in a preconditioner setting by the coarsening factors used, and the degree of smoothing applied to the basis. We also compare with other state-of-the-art AMG implementations, but do not investigate combinations of them with the MSRB method. For the MsRSB method, we also discuss practical considerations in different parallelization regimes including domain decomposition using MPI, shared memory using OpenMP, and GPU acceleration with CUDA.
All comparisons will focus on the setting in which many similar systems should be solved, e.g. during a large-scale, multiphase flow simulation. That is, our emphasis is on the performance of updating a preconditioner and on the apply time for the preconditioner relative to the convergence rate. Performance of the solvers will be tested for pure parabolic/elliptic problems that either arise as part of a sequential splitting procedure or as a pseudo-elliptic preconditioner/solver as a part of a CPR preconditioner for a multiphase system, for which block ILU0 is used as the outer smoother.
So far, most work on comparing MsRSB with AMG methods has been on qualitative performance measures like iteration number rather than on pure runtime on fair code implementation. We discuss the theoretical performance and show the practical performance for our implementation. Here, we compare performance of pure AMG, standard two-level MsRSB with pure AMG as coarse solver, as well as a new truly multilevel MsRSB scheme. Our implementation uses the DUNE-ISTL framework. To limit the scope of the discussion we restrict our assessment to AMG with aggregation and smoothed aggregation and the MsRSB method. These three methods are closely related and are primarily distinguished in a preconditioner setting by the coarsening factors used, and the degree of smoothing applied to the basis. We also compare with other state-of-the-art AMG implementations, but do not investigate combinations of them with the MSRB method. For the MsRSB method, we also discuss practical considerations in different parallelization regimes including domain decomposition using MPI, shared memory using OpenMP, and GPU acceleration with CUDA.
All comparisons will focus on the setting in which many similar systems should be solved, e.g. during a large-scale, multiphase flow simulation. That is, our emphasis is on the performance of updating a preconditioner and on the apply time for the preconditioner relative to the convergence rate. Performance of the solvers will be tested for pure parabolic/elliptic problems that either arise as part of a sequential splitting procedure or as a pseudo-elliptic preconditioner/solver as a part of a CPR preconditioner for a multiphase system, for which block ILU0 is used as the outer smoother.