Development of a deflation-based linear solver in reservoir simulation

Development of a deflation-based linear solver in reservoir simulation

Van der Linden, J.H.

Jonsthovel, T.B. (mentor)
Vuik, C. (mentor)

Electrical Engineering, Mathematics and Computer Science

Numerical Analysis

2013-12-02

Extreme and isolated eigenvalues are known to be harmful to the convergence of the iterative solver. These eigenvalues are produced by strong heterogeneity in the underlying physics. We can improve the quality of the spectrum by ‘deflating’ the harmful eigenvalues. In this thesis, deflation is applied to linear systems in reservoir simulation. We show that large, sudden differences in the permeability produce extreme eigenvalues. For small cases, the number and magnitude of these eigenvalues is linked to the number and magnitude of the permeability jumps. Two deflation methods are investigated. Firstly, we show that harmonic Ritz eigenvector deflation, which computes the deflation vectors from the information produced by the linear solver, can improve convergence. The computational cost of this method, however, is relatively high and the method cannot be implemented in parallel. Secondly, we test a physics-based deflation algorithm that constructs the deflation vectors a priori. The method is shown to improve the performance of the linear solver. We compare manually constructed deflation vectors to a partitioner algorithm, which automatically identifies large permeability jumps and constructs the deflation vectors using the subdomain-levelset method. Automatic (parallel) physics-based deflation works well for small cases, but we also show that the partitioner algorithm is not robust in large, realistic cases. We make several suggestions for improvement, such as assigning deflation vectors only to regions where flow occurs. For cases with well-defined permeability jumps of a factor 10^4 or higher, we believe that physics-based deflation has a large potential.

deflation
linear solver
preconditioning
GMRES
CPR
Harmonic Ritz
subdomain levelset
porous media flow

http://resolver.tudelft.nl/uuid:47cbb291-6b1e-4572-b384-f79a8cf7e535

Student theses

master thesis

(c) 2013 Van der Linden, J.H.