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Asynchronous Multirate Newton - A Class of Nonlinear Solver that Adaptively Localizes Computation
- Publisher: European Association of Geoscientists & Engineers
- Source: Conference Proceedings, ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery, Sep 2014, Volume 2014, p.1 - 16
Abstract
Summary
Locality is inherent to all transient flow and transport phenomena. Moreover, the superposition of the two disparate spatiotemporal scales that underlie flow and transport leads to a problem of dimensionality. Numerous Adaptive discretization methods have been devised to exploit an a priori understanding of locality. While such methods have provided various degrees of success, they are fundamentally restricted by the fidelity of the discretization under aggressive adaptivity. This work seeks a novel class of nonlinear solvers which are proposed to adapt the level of computation to precisely match that of the underlying spatiotemporal change, without affecting the underlying discretization model. We devise a class of nonlinear iteration that on the one hand, converges as rapidly as the best available safeguarding method (e. g. trust-region), while on the other, performing a number of operations that is at most equal to the number of cells that experience changes over an iteration.
We achieve this by developing an Asynchronous Multirate numerical integration of the Newton Flow differential system. At each nonlinear iteration, the domain of interest is partitioned adaptively in two or more levels of disjoint subdomains on the basis of the predicted rate of change of state variables. On one extreme, there are only two levels of partitions; cells that will experience a nonzero change, and cells will not. This two-level solver is equivalent to the adaptively localized solution of the linear Jacobian system. On the other extreme, the domain is decomposed into multiple disjoint subdomains. Under this scheme, the subdomains are solved sequentially using a dynamic partitioning strategy in the order from fastest to slowest. We present detailed computational results focused on general multiphase flow models. The performance improvement directly depends on the extent of locality present in the model. On the two-level end of the spectrum, the convergence rate of the proposed method is unadulterated while the performance is improved by an order of magnitude in computational time.
On the multilevel end of the spectrum, while additional performance gains are obtained for transport components, the convergence rate for pressure requires more costly synchronization strategies. This method looks very promising for the simulation of extremely complex models where well controls change dynamically.
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