Multigrid Methods with Applications to Reservoir Simulation

Multigrid Methods with Applications to Reservoir Simulation
Author:
Publisher:
Total Pages: 246
Release: 1994
Genre:
ISBN:
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Multigrid methods are studied for solving elliptic partial differential equations. Focus is on parallel multigrid methods and their use for reservoir simulation. Multicolor Fourier analysis is used to analyze the behavior of standard multigrid methods for problems in one and two dimensions. Relation between multicolor and standard Fourier analysis is established. Multiple coarse grid methods for solving model problems in 1 and 2 dimensions are considered; at each coarse grid level we use more than one coarse grid to improve convergence. For a given Dirichlet problem, a related extended problem is first constructed; a purification procedure can be used to obtain Moore-Penrose solutions of the singular systems encountered. For solving anisotropic equations, semicoarsening and line smoothing techniques are used with multiple coarse grid methods to improve convergence. Two-level convergence factors are estimated using multicolor. In the case where each operator has the same stencil on each grid point on one level, exact multilevel convergence factors can be obtained. For solving partial differential equations with discontinuous coefficients, interpolation and restriction operators should include information about the equation coefficients. Matrix-dependent interpolation and restriction operators based on the Schur complement can be used in nonsymmetric cases. A semicoarsening multigrid solver with these operators is used in UTCOMP, a 3-D, multiphase, multicomponent, compositional reservoir simulator. The numerical experiments are carried out on different computing systems. Results indicate that the multigrid methods are promising.

Multigrid Methods for Process Simulation

Multigrid Methods for Process Simulation
Author: Wolfgang Joppich
Publisher: Springer Science & Business Media
Total Pages: 327
Release: 2012-12-06
Genre: Technology & Engineering
ISBN: 3709192536
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It was about 1985 when both of the authors started their work using multigrid methods for process simulation problems. This happened in dependent from each other, with a completely different background and different intentions in mind. At this time, some important monographs appeared or have been in preparation. There are the three "classical" ones, from our point of view: the so-called "1984 Guide" [12J by Brandt, the "Multi-Grid Methods and Applications" [49J by Hackbusch and the so-called "Fundamentals" [132J by Stiiben and Trottenberg. Stiiben and Trottenberg in [132J state a "delayed acceptance, resent ments" with respect to multigrid algorithms. They complain: "Nevertheless, even today's situation is still unsatisfactory in several respects. If this is true for the development of standard methods, it applies all the more to the area of really difficult, complex applications." In spite of all the above mentioned publications and without ignoring important theoretical and practical improvements of multigrid, this situa tion has not yet changed dramatically. This statement is made under the condition that a numerical principle like multigrid is "accepted", if there exist "professional" programs for research and production purposes. "Professional" in this context stands for "solving complex technical prob lems in an industrial environment by a large community of users". Such a use demands not only for fast solution methods but also requires a high robustness with respect to the physical parameters of the problem.

Multigrid Methods IV

Multigrid Methods IV
Author: P.W. Hemker
Publisher: Birkhäuser
Total Pages: 360
Release: 2012-12-06
Genre: Mathematics
ISBN: 3034885245
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This volume contains a selection from the papers presented at the Fourth European Multigrid Conference, held in Amsterdam, July 6-9,1993. There were 78 registered participants from 14 different countries, and 56 presentations were given. The preceding conferences in this series were held in Cologne (1981, 1985) and in Bonn (1990). Also at the other side of the Atlantic special multigrid conferences are held regularly, at intervals of two years, always in Copper Mountain, Colorado, US. The Sixth Copper Mountain Conference on Multigrid Methods took place in April, 1993. Circumstances prevented us from putting a larger time interval between the Copper and Amsterdam meetings. The next European meeting is planned in 1996, a year later than the next Copper Meeting. When the first multigrid conference was held in 1981 there was no doubt about the usefulness of a conference dedicated specially to multigrid, because multigrid was a new and relatively unexplored subject, still in a pioneering stage, and pursued by specialists. The past twenty years have shown a rapid growth in theoretical understanding, useful applications and widespread acceptance of multi grid in the applied disciplines. Hence, one might ask whether there is still a need today for conferences specially dedicated to multigrid. The general consensus is that the answer is affirmative. New issues have arisen that are best addressed or need also be addressed from a special multigrid point of view.

Multigrid Methods

Multigrid Methods
Author: Stephen F. McCormick
Publisher: SIAM
Total Pages: 292
Release: 1987-12-01
Genre: Mathematics
ISBN: 1611971888
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A thoughtful consideration of the current level of development of multigrid methods, this volume is a carefully edited collection of papers that addresses its topic on several levels. The first three chapters orient the reader who is familiar with standard numerical techniques to multigrid methods, first by discussing multigrid in the context of standard techniques, second by detailing the mechanics of use of the method, and third by applying the basic method to some current problems in fluid dynamics. The fourth chapter provides a unified development, complete with theory, of algebraic multigrid (AMG), which is a linear equation solver based on multigrid principles. The last chapter is an ambitious development of a very general theory of multigrid methods for variationally posed problems. Included as an appendix is the latest edition of the Multigrid Bibliography, an attempted compilation of all existing research publications on multigrid.

Multigrid Methods for Process Simulation

Multigrid Methods for Process Simulation
Author: Wolfgang Joppich
Publisher: Springer
Total Pages: 309
Release: 1993-08-20
Genre: Technology & Engineering
ISBN: 9783211824047
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It was about 1985 when both of the authors started their work using multigrid methods for process simulation problems. This happened in dependent from each other, with a completely different background and different intentions in mind. At this time, some important monographs appeared or have been in preparation. There are the three "classical" ones, from our point of view: the so-called "1984 Guide" [12J by Brandt, the "Multi-Grid Methods and Applications" [49J by Hackbusch and the so-called "Fundamentals" [132J by Stiiben and Trottenberg. Stiiben and Trottenberg in [132J state a "delayed acceptance, resent ments" with respect to multigrid algorithms. They complain: "Nevertheless, even today's situation is still unsatisfactory in several respects. If this is true for the development of standard methods, it applies all the more to the area of really difficult, complex applications." In spite of all the above mentioned publications and without ignoring important theoretical and practical improvements of multigrid, this situa tion has not yet changed dramatically. This statement is made under the condition that a numerical principle like multigrid is "accepted", if there exist "professional" programs for research and production purposes. "Professional" in this context stands for "solving complex technical prob lems in an industrial environment by a large community of users". Such a use demands not only for fast solution methods but also requires a high robustness with respect to the physical parameters of the problem.