A characteristic nonoverlapping domain decomposition method for multidimensional convection‐diffusion equations

We develop a quasi‐two‐level, coarse‐mesh‐free characteristic nonoverlapping domain decomposition method for unsteady‐state convection‐diffusion partial differential equations in multidimensional spaces. The development of the domain decomposition method is carried out by utilizing an additive Schwarz domain decomposition preconditioner, by using an Eulerian‐Lagrangian method for convection‐diffusion equations and by delicately choosing appropriate interface conditions that fully respect and utilize the hyperbolic nature of the governing equations. Numerical experiments are presented to illustrate the method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An improved numerical simulator for multiphase flow in porous media

In basin modelling the thermodynamics of a multicomponent multiphase fluid flux are computationally too expensive when derived from an equation of state and the Gibbs equality constraints. In this article we present a novel implicit molar mass formulation technique using binary mixture thermodynamics. The two proposed solution methods, with and without cross derivative terms between components, are based on a preconditioned Newton‐GMRES scheme for each time‐step with analytical computation of the derivatives. These new algorithms reduce significantly the numerical effort for the computation of the molar masses, and we illustrate the behavior of these methods with numerical computations. Copyright © 2004 John Wiley & Sons Ltd.     

Convergent iterative schemes for time parallelization

Parallel methods are usually not applied to the time domain because of the inherit sequentialness of time evolution. But for many evolutionary problems, computer simulation can benefit substantially from time parallelization methods. In this paper, we present several such algorithms that actually exploit the sequential nature of time evolution through a predictor-corrector procedure. This sequentialness ensures convergence of a parallel predictor-corrector scheme within a fixed number of iterations. The performance of these novel algorithms, which are derived from the classical alternating Schwarz method, are illustrated through several numerical examples using the reservoir simulator Athena.

     

Wavelet upscaling based on piecewise bilinear approximation of the permeability field

A method for upscaling of permeability in heterogeneous porous media is presented. The upscaled field takes the form K = e ^Y , where Y, in two dimensions, is a piecewise bilinear function. The method is tested on a number of random permeability fields, with different integral scale/correlation length and variance. The numerical results show that this method conserves much more of the heterogeneous fingering than classical block-based upscaling methods, e.g., geometric mean.     

A hybrid nonoverlapping domain decomposition scheme for advection dominated advection–diffusion problems

Two nonoverlapping domain decomposition algorithms are proposed for convection dominated convection–diffusion problems. In each subdomain, artificial boundary conditions are used on the inflow and outflow boundaries. If the flow is simple, each subdomain problem only needs to be solved once. If there are closed streamlines, an iterative algorithm is needed and the convergence is proved. Analysis and numerical tests reveal that the methods are advantageous when the diffusion parameter ɛ is small. In such cases, the error introduced by the domain decomposition methods is negligible in comparison with the error in the singular layers, and it allows easy and efficient grid refinement in the singular layers. This revised version was published online in August 2006 with corrections to the Cover Date.