On the Convergence of Nonlinear Finite Volume Methods

Despite being the industry standard for many years, the two-point flux approximation (TPFA) is only consistent on K-orthogonal grids and might show significant grid orientation errors for tensorial permeability and general cell geometries. Nonlinear flux discretizations, while capable of generating consistent and monotone finite volume (FV) schemes, introduce added computational complexity due to scheme nonlinearity, even when applied to linear partial differential equations (PDEs). Although some studies have identified nonlinear schemes that can potentially outperform multi-point flux approximation (MPFA) methods in terms of computational efficiency, the inherent nonlinearity poses an additional layer of complexity compared to linear discretizations. The efficiency of nonlinear finite volume (FV) computations is influenced by the selection of nonlinear and linear solvers, along with the specified stopping criteria (i.e., tolerance) for each solver. This, in turn, impacts the overall number of iterations and the computational cost per iteration. This work focuses on examining the convergence behavior of nonlinear finite volume methods, with particular attention given to the choice of linear solvers. In proof-of-concept studies, linearized systems were solved exactly using direct methods or to very tight tolerances using iterative methods, aiming to establish the non-negativity of solutions throughout all iterations. For the elliptic pressure equation linearized with Picard iterations, we show that using linear solvers to approximate the solution of the linear system for the first few nonlinear iterations can significantly reduce the total number of linear iterations. The efficacy of this proposed solution approach is assessed through the utilization of various linear solvers, including Bi-Conjugate Gradient Stabilized (BiCGStab), Generalized Minimal Residual (GMRES), Multiscale Restriction Smoothed Basis (MsRSB), and Multigrid solvers. For example, when employing the MsRSB solver, our results demonstrate that it is possible to maintain nearly the same number of nonlinear iterations while significantly reducing the overall CPU time-almost halving the time required when the same solver is used to solve all linearized systems to a tight tolerance. This solution strategy is further explored in the realm of multiphase flow simulations using the sequential fully implicit (SFI) scheme. To the best of our knowledge, nonlinear discretizations are yet to be used in simulations using a SFI scheme. Furthermore, for the multiphase flow problems using the SFI scheme, we investigate the performance of Newton's method as a nonlinear solver as opposed to using Picard's method.