Incomplete Gaussian Elimination as a Preconditioning for Generalized Conjugate Gradient Acceleration

Wallis, J.R.

doi:10.2118/12265-ms

Cited by 151 publications

(50 citation statements)

References 14 publications

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“…Because we work with a symmetric structure, the pattern of fill-in is also symmetric. Following Wallis, 8 for each fill-in term xij in the (i,j) position of the matrix, a level £(x i j) is associated. Initially define…”

Section: Incomplete Factorizationmentioning

confidence: 99%

Incomplete LU Preconditioners for Conjugate-Gradient-Type Iterative Methods

Simon

1988

SPE Reservoir Engineering

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In receht years, such conjugate-gradient-type methods as orthomin have been used very successfully with various preconditioners to solve the unsymmetric linear systems that arise in reservoir simulation. Here these successful iterative methods are combined with a new set of preconditioners that have been derived from some powerful algorithms in direct sparse Gaussian elimination. The SPARSPAK implementations of minimum degree and nested dissection algorithms have been modified to yield incomplete LV factorizations. Their application to reservoir simulatioh problems is investigated here.

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Section: Incomplete Factorizationmentioning

confidence: 99%

Incomplete LU Preconditioners for Conjugate-Gradient-Type Iterative Methods

Simon

1988

SPE Reservoir Engineering

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“…After the size of the system is reduced, the resulting global linear system of equations is solved for the primary variables. The linear solver employs GMRES and for preconditioning we use the two-stage Constrained Pressure Residual (CPR) strategy presented in Wallis (1983) and Wallis (1983). An algebraic reduction step is used to construct the pressure equation in the first stage, which is then solved using an AMG solver (Stueben, 2007).…”

Section: Solution Methodsmentioning

confidence: 99%

General Nonlinear Solution Strategies for Multiphase Multicomponent EoS Based Simulation

Voskov

Younis

2009

All Days

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We present a novel reservoir flow simulation framework built on top of an Automatic Differentiation with Expression Templates Library (ADETL). Given discrete algebraic expressions of the governing conservation equations, associated constraint relations, and equations of state, the ADETL library generates efficient computer code for the corresponding Jacobian matrix. The framework allows for complete flexibility in the specification of the independent variables, which leads to unified simulator for different formulations and solution strategies. We study the behavior of several widely used nonlinear formulations and solution strategies for general-purpose compositional simulation. The numerical behaviors using different variable sets, including the so called natural-and mass-type variables, are investigated. We also report on tie-line based parameterization strategies for compositional modeling. The focus is on the fully implicit method (FIM), where the full Jacobian (primary and secondary algebraic equations in terms of both the primary and secondary variables) is constructed. Schur-complement reduction of the full Jacobian is used to assemble the (reduced) Jacobian for the primary equations as a function of the primary unknowns only. The numerical challenges associated with the nonlinear hydrodynamic and thermodynamic interactions associated with compositional displacement processes are the focus of this investigation. These include phase appearance and disappearance, crossing into and out of the critical region, wide pressure variations in space and time. The behavior of the Newton iterations as a function of timestep size is analyzed for different variable sets, alignment of equations (including linear and nonlinear constraints), and nonlinear updating strategies. Comparisons are performed for several challenging multi-phase multi-component problems of practical interest.

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“…Although preconditioned orthomin has been shown to be convergent on in-situ combustion problems, 13,14 none was used in this study.…”

Section: Test Problemsmentioning

confidence: 99%

Application of Preconditioned Conjugate-Gradient-Like Methods in Reservoir Simulation

Northrup

Woo

1988

SPE Reservoir Engineering

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·Conjugate-gradient-like (CGL) methods were used to solve the matrix equations for several large-field and fine-grid models. The methods used incomplete LU (lLU) or modified incomplete LU (MILU) preconditioning coupled with orthomin acceleration. The CGL methods were applied to the original matrix or to a reduced system from red/black ordering. Test results from a black-oil model with fault throws, a fractured reservoir model, and three steamflood models are given. It was found that the CGL methods are effective for solving the matrix equations. A more powerful preconditioner improves reliability but may require more work. Guidelines for selecting the most effective method are provided.

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Incomplete Gaussian Elimination as a Preconditioning for Generalized Conjugate Gradient Acceleration

Abstract: This paper discusses the application of preconditioned generalized conjugate gradient acceleration to fully implicit thermal simulation. The preconditioning step utilizes incomplete Gaussian elimination (IGE) to form an approximate factorization of the Jacobian matrix.

Cited by 151 publications

References 14 publications

Incomplete LU Preconditioners for Conjugate-Gradient-Type Iterative Methods

Incomplete LU Preconditioners for Conjugate-Gradient-Type Iterative Methods

General Nonlinear Solution Strategies for Multiphase Multicomponent EoS Based Simulation

Application of Preconditioned Conjugate-Gradient-Like Methods in Reservoir Simulation

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