Gunnar E. Fladmark

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.

A Convergent Algorithm for Time Parallelization Applied to Reservoir Simulation

Parallel methods are not usually applied to the time domain because the sequential nature of time is considered to be a handicap for the development of competitive algorithms. However, this sequential nature can also play to our advantage by ensuring convergence within a given number of iterations. The novel parallel algorithm presented in this paper acts as a predictor corrector improving both speed and accuracy with respect to the sequential solvers. Experiments using our in house fluid flow simulator in porous media, Athena, show that our parallel implementation exhibit an optimal speed up relative to the method.

Which meshes are better conditioned: adaptive, uniform, locally refined or locally adjusted?

Adaptive, locally refined and locally adjusted meshes are preferred over uniform meshes for capturing singular or localised solutions. Roughly speaking, for a given degree of freedom a solution associated with adaptive, locally refined and locally adjusted meshes is more accurate than the solution given by uniform meshes. In this work, we answer the question which meshes are better conditioned. We found, for approximately same degree of freedom (same size of matrix), it is easier to solve a system of equations associated with an adaptive mesh.

Parallelization of a Compositional Reservoir Simulator

A finite volume dicretization has been used to solve compositional flow in porous media. Secondary migration in fractured rocks has been the main motivation for the work. Multipoint flux approximation has been implemented and adaptive local grid refinement, based on domain decomposition, is used at fractures and faults. The parallelization method, which is described in this paper, strongly promotes code reuse and gives a very high level of parallelization despite low implementation costs. The programming framework is also portable to other platforms or other applications. We have presented computer experiments to examine the parallel efficiency of the implemented parallel simulator with respect to scalability and speedup.

Control Volume Finite Difference On Adaptive Meshes

In this work we present a finite volume discretization of an elliptic boundary value problem on adaptively refined meshes. This problem is important in many practical applications, e.g. porous media flow. We propose an error indicator functional which is used to select elements that should be refined. Two numerical examples are provided to demonstrate the potential of the proposed refinement strategy.

A Preconditioning Technique as an Upscaling Procedure

A preconditioning technique has been used as an upscaling method, Numerical results created by the preconditioning technique are compared with numerical results created by a homogenization technique. This is done in 2D for both one- and two-phase flow problems. Large differences , which favor the preconditioning technique, are observed for two-phase flow problems. The method is also implemented and tested in 3D.