Solver preconditioning using the combinatorial multilevel method

Yuhe Wang, John E. Killough

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The purpose of this paper is to report the first preliminary study of the recently introduced Combinatorial Multilevel (CML) method for solver preconditioning in large-scale reservoir simulation with coupled geomechanics. The CML method is a variant of the popular Algebraic Multigrid (AMG) method yet with essential differences. The basic idea of this new approach is to construct a hierarchy of matrices by viewing the underlying matrix as a graph and by using the discrete geometry of the graph such as graph separators and expansion. In this way, the CML method combines the merits of both geometric and algebraic multigrid methods. The resulting hybrid approach not only provides a simpler and faster set-up phase compared to AMG, but the method can be proved to exhibit strong convergence guarantees for arbitrary symmetric diagonally-dominant matrices. In addition, the underlying theoretical soundness of the CML method contrasts to the heuristic AMG approach, which often can show slow convergence for difficult problems. This new approach is implemented in a reservoir simulator for both pressure and displacement preconditioners in the multi-stage preconditioning technique. We present results based on several known benchmark problems and provide a comparison of performance and complexity with the widespread preconditioning schemes used in large-scale reservoir simulation. An adaptation of CML for unsymmetric matrices is shown to exhibit excellent convergence properties for realistic cases.

Original languageEnglish
Title of host publicationSociety of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013
Pages182-196
Number of pages15
Volume1
Publication statusPublished - 2013
EventSPE Reservoir Simulation Symposium 2013 - The Woodlands, TX, United States
Duration: 18 Feb 201320 Feb 2013

Other

OtherSPE Reservoir Simulation Symposium 2013
CountryUnited States
CityThe Woodlands, TX
Period18/2/1320/2/13

Fingerprint

Multilevel Methods
Preconditioning
Algebraic multigrid Method
Reservoir Simulation
Algebraic multigrid
matrix
Geomechanics
Graph in graph theory
Diagonally Dominant Matrix
Discrete Geometry
Preconditioning Techniques
Separators
Separator
Hybrid Approach
Soundness
Strong Convergence
Symmetric matrix
Preconditioner
Simulators
Convergence Properties

ASJC Scopus subject areas

  • Geochemistry and Petrology
  • Modelling and Simulation

Cite this

Wang, Y., & Killough, J. E. (2013). Solver preconditioning using the combinatorial multilevel method. In Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013 (Vol. 1, pp. 182-196)

Solver preconditioning using the combinatorial multilevel method. / Wang, Yuhe; Killough, John E.

Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013. Vol. 1 2013. p. 182-196.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Wang, Y & Killough, JE 2013, Solver preconditioning using the combinatorial multilevel method. in Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013. vol. 1, pp. 182-196, SPE Reservoir Simulation Symposium 2013, The Woodlands, TX, United States, 18/2/13.
Wang Y, Killough JE. Solver preconditioning using the combinatorial multilevel method. In Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013. Vol. 1. 2013. p. 182-196
Wang, Yuhe ; Killough, John E. / Solver preconditioning using the combinatorial multilevel method. Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013. Vol. 1 2013. pp. 182-196
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