Multi-level adaptive solutions to boundary-value problems
01 Apr 1977-Mathematics of Computation (American Mathematical Society (AMS))-Vol. 31, Iss: 138, pp 333-390
TL;DR: In this paper, the boundary value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, and interactions between these levels enable us to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); and conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.
Abstract: The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining \"°°-order\" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly
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TL;DR: The vorticity-stream function formulation of the two-dimensional incompressible NavierStokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions.
4,018 citations
01 May 1995
TL;DR: This paper quantitatively characterize the SPLASH-2 programs in terms of fundamental properties and architectural interactions that are important to understand them well, including the computational load balance, communication to computation ratio and traffic needs, important working set sizes, and issues related to spatial locality.
Abstract: The SPLASH-2 suite of parallel applications has recently been released to facilitate the study of centralized and distributed shared-address-space multiprocessors. In this context, this paper has two goals. One is to quantitatively characterize the SPLASH-2 programs in terms of fundamental properties and architectural interactions that are important to understand them well. The properties we study include the computational load balance, communication to computation ratio and traffic needs, important working set sizes, and issues related to spatial locality, as well as how these properties scale with problem size and the number of processors. The other, related goal is methodological: to assist people who will use the programs in architectural evaluations to prune the space of application and machine parameters in an informed and meaningful way. For example, by characterizing the working sets of the applications, we describe which operating points in terms of cache size and problem size are representative of realistic situations, which are not, and which re redundant. Using SPLASH-2 as an example, we hope to convey the importance of understanding the interplay of problem size, number of processors, and working sets in designing experiments and interpreting their results.
4,002 citations
Book•
01 Jan 1982TL;DR: This work presents an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques based upon Richardson-type estimates of the truncation error, which is a mesh refinement algorithm in time and space.
Abstract: We present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Our approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, data structures and grid generation procedure, and conclude with numerical examples in one and two space dimensions.
2,120 citations
TL;DR: The aim of this paper is to present the reader with a perspective on how JFNK may be applicable to applications of interest and to provide sources of further practical information.
1,803 citations
Book•
01 Jan 1992
TL;DR: These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians, restricting ourselves to finite volume and finite difference discretization.
Abstract: These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians. The use of more advanced mathematical tools, such as functional analysis, is avoided. The course is intended to be accessible to a wide audience of users of computational methods. We restrict ourselves to finite volume and finite difference discretization. The basic principles are given. Smoothing methods and Fourier smoothing analysis are reviewed. The fundamental multigrid algorithm is studied. The smoothing and coarse grid approximation properties are discussed. Multigrid schedules and structured programming of multigrid algorithms are treated. Robustness and efficiency are considered.
1,291 citations
References
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01 Sep 1971
4,078 citations
01 Jan 1967
TL;DR: In this article, differentielles and stabilite were used for differentiable transport in the context of transfert de chaleur and ondes Reference Record created on 2005-11-18, modified on 2016-08-08
Abstract: Keywords: equations : differentielles ; stabilite ; transport ; transfert de chaleur ; mecanique des : fluides ; ondes Reference Record created on 2005-11-18, modified on 2016-08-08
3,449 citations
TL;DR: The present article describes a method that is developed for iterative solution of elliptic difference equations, using an idea not unlike the relaxation method, which was put into practice on a digital computer and gave good results.
Abstract: WHEN solving elliptic equations by the method of fmite differences we have to deal with systems of linear algebraic equations, often of a very high order. Given a sufficiently high order of the system, the familiar iterative methods of solution of such systems are very slowly convergent. Numerous works have been devoted to methods of speeding up the convergence of the iterations. These speedingup methods can be split provisionally into two groups. The first group includes methods which use the spectrum of the iterative operators; they are described in detail in text-books [l] and [2]. The second, rather indefinite group includes the so-called “relaxation” methods, that are based essentially on “intuition”, the “computer’s experience”; they are regarded as applicable for nonmechanical computation by a sufficiently experienced group of workers, but as little suited to being carried out on digital computers; it is usually suggested that the relaxation method can be extremely effective [3]. The present article describes a method that we have developed for iterative solution of elliptic difference equations, using an idea not unlike the relaxation method. The present method was put into practice on a digital computer and gave good results. Le us take Poisson’s equation in a rectangular domain:
308 citations
297 citations
TL;DR: In this paper, the requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies, and sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator.
Abstract: The requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies. A mathematical analysis shows that sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator. The latter results in a consistency condition on the integral equations, rather than the differential, at these points. Calculated jump conditions for several embedded and detached shock waves are analyzed in the physical and hodograph planes. Comparisons are made with previous results, a time-dependent calculation, and data.
261 citations