Abstract
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.
Highlights
Solutions of systems of linear equations are required when simulating flow through subsurface porous media [27]
In this work, we study the performance of the Deflated Preconditioned Conjugate Gradient method preconditioned with Incomplete Cholesky (DICCG) and a Proper Orthogonal Decomposition (POD) basis as deflation vectors
We study two different approaches: Moving Window (MW) and Training Phase (TP), that differ in they way the snapshots are collected
Summary
Solutions of systems of linear equations are required when simulating flow through subsurface porous media [27]. These linear systems emerge during the iterative solution of time- and space-discretized nonlinear partial differential equations that govern the porous media flow problems. For the IMPES scheme, the saturation is calculated explicitly, while pressure is kept implicit, and solving the pressure equation is the most time-consuming part, especially for large and ill-conditioned systems, which often occur because of the complex geometry and strongly heterogeneous rock properties that constitute the porous medium. For time-varying problems, it is required to compute a large number of simulations, which makes the solution of these problems expensive
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