The alternate-block-factorization procedure for systems of partial differential equations

Abstract

Abstract.The alternate-block-factorization (ABF) method is a procedure for partially decoupling systems of elliptic partial differential equations by means of a carefully chosen change of variables. By decoupling we mean that the ABF strategy attempts to reduce intra-equation coupling in the system rather than intra-grid coupling for a single elliptic equation in the system. This has the effect of speeding convergence of commonly used iteration schemes, which use the solution of a sequence of linear elliptic PDEs as their main computational step. Algebraically, the change of variables is equivalent to a postconditioning of the original system. The results of using ABe: postconditioning on some problems arising from semiconductor device simulation are discussed.

Physics-Based Preconditioning and the Newton–Krylov Method for Non-equilibrium Radiation Diffusion

“…Here the constant k is varied in Eq. (17) to show the effect that material conduction has on the solution. It is clear that an increase in the material diffusion coefficient causes the thermal front to move faster.…”

“…In the early days of computational physics the numerical technique of operator splitting [2] (or fractional time step methods [16] or alternate-block factorization [17]) was the main work horse for the solution of coupled systems. This approach reduces the memory requirements and the computational complexity of solving a system of equations.…”

Algebraic Multigrid Methods (AMG) for the Efficient Solution of Fully Implicit Formulations in Reservoir Simulation

“…Therefore, we can expect that the degree of coupling of the off-diagonal blocks has been reduced to some extent This decoupling operator was proposed by Bank et al 27 as the Alternate-Block Factorization (ABF) procedure and is subject to further analysis in other works 28,29 . The ABF operator was successfully applied by Klie 13 for two-phase flow problems where Properties 1-5 from above were shown to be strongly satisfied under mild conditions (e.g., if blocks ps A and sp A satisfy the M-matrix condition).…”

A multi-continuum multi-phase parallel simulator for large-scale conventional and unconventional reservoirs

“…If only oil phase exists, X i ,i ¼ 1,…,nÀ1 are the component mole fractions in oil phase x i ,i ¼ 1,…,nÀ1; If only gas phase exists, X i ,i ¼ 1,…,nÀ1 are the component mole fractions in gas phase y i ,i ¼ 1,…,nÀ1; If oil and gas phases coexist, X 1 is the gas phase saturation, and X i ,i ¼ 2,…,nÀ1 are the component mole fraction of gas phase y i ,i ¼ 1,…,nÀ2. The alternative block factorization (ABF) decoupling method (Bank et al, 1989) used as the preprocessing step to weaken the strong nonlinear coupling between the pressure unknowns and other unknowns in compositional model can be written as…”