Mathematics, Parallel Computing and Reservoir Simulation

This paper will highlight, by way of examples, a few seemingly very different mathematical problems and show how they have direct relevance to the construction of efficient computational procedures for the simulation of oil reservoirs on parallel computers. 1 Partitioning of graphs Consider a graph G(V; E), having a set of vertices connected by edges. We associate work with each vertex and communication (or dependencies) with each edge. Our problem is to partition the graph into two parts such that the two sets are similar in size. We further want the dependencies (i.e., the number of edges) between the two sets to be as small as possible. Mathematically this can be expressed by having a variable x i at vertex V i be one or minus one depending on which of the two sets V i belongs to. A little reflection shows that our problem can be formulated min X E ij (x i \Gamma x j ) 2 subject to X i x i = 0; x i = \Sigma1: Define the degree d i of vertex V i as the number of edges conne..

Parallel Domain Decomposition And Iterative Refinement Algorithms

. Algorithms for the solution of partial differential equations based on a subdivision of the spatial domain, has received much interest in recent years. To a large extent this has been motivated by the new generation of parallel computers. This algorithmic approach can introduce independent parallel tasks of variable granularity, depending on the subdivision and can therefore be adapted to a wide range of parallel computers. We review some of the progress that has been made and supply a few new numerical examples that illustrate the convergence behavior. Key Words. Schwarz' Method, Iterative Substructuring Methods, Domain Decomposition, Elliptic Equations, Finite Elements, Iterative Refinement, Parallel computing. 1. Introduction. Recently there has been a strong revival of the interest in domain decomposition algorithms for elliptic problems; cf. e.g. Glowinski et al, [17], and Chan et al, [10]. This is to a large extent due to their potential on multiprocessor computing systems. It..

Multiplicative And Additive Schwarz' Methods: Convergence In The 2-Domain Case.

. We consider the classical Schwarz alternating algorithm and an additive version more suitable for parallel processing. The two methods are compared and analyzed in the case of two domains. We show that the rate of convergence for both methods, can be directly related to a generalized eigenvalue problem, derived from subdomain contributions to the global stiffness matrix. Analytical expressions are given for a model case. Key Words. Schwarz' Method, Orthogonal Projections, Elliptic Equations, Domain Decomposition, Finite Elements, Parallel Computation. 1. Introduction. Recently there has been a strong revival of the interest in domain decomposition algorithms for the solution of elliptic problems; cf. e.g. Glowinski et al [12], and Chan et al [7]. This is to a large extent due to their potential in parallel computing environments. Substructuring methods, with a long history from the structural analysis community [20, 1], are methods where the global domain is partitioned into disjoin..

Efficient Algorithms For Solving A Fourth Order Equation With The Spectral-Galerkin Method

. We show that one can derive an O(N 3 ) spectral-Galerkin method for fourth order (biharmonic type) elliptic equations based on the use of Chebyshev polynomials. The use of Chebyshev polynomials provides a fast transform between physical and spectral space which is advantageous when a sequence of problems must be solved e.g., as part of a nonlinear iteration. This improves the result of Shen [9] which reported an O(N 4 ) algorithm inferior to the O(N 3 ) method developed earlier [8] based on Legendre polynomials, but less practical in the case of multiple problems. We further compare our method with an improved implementation of the Legendre-Galerkin method based on the same approach. Key words. spectral-Galerkin method, Chebyshev polynomial, Legendre polynomial, Helmholtz equation, biharmonic equation, direct solver, iterative solver. AMS(MOS) subject classification. 65N35, 65N22, 65F05, 35J05. 1. Introduction. In his recent work [8] and [9] Shen develops a class of spectral..

A Coarse Space Formulation with good parallel properties for an Additive Schwarz Domain Decomposition Algorithm

This paper studies a variant of the Additive Average Schwarz algorithm [1] where the coarse space consists of two subspaces. This approach results in a simpler and more parallel algorithm while we retain the essential convergence properties of the original method. Our theory is confirmed by numerical experiments showing that the new algorithm is often superior to the original variant. 1 Introduction In [1] two additive Schwarz algorithms were described and analyzed. The first, called the Additive Diagonal Scaling method, used a diagonal scaling derived from a coarse space V \Gamma1 (see below), combined with the classical coarse space V c 0 . This method was not robust when the jumps in the coefficients of the PDE could be arbitrarily distributed across subdomains. The second method, called the Additive Average Schwarz method, was proven robust, but uses a potentially expensive coarse space V avg 0 = Range(IA ) with I Au = ae u(x); x 2 @\Omega ih ¯ u i ; x 2\Omega ih ; ..

Domain Decomposition, Parallel Computing and Petroleum Engineering

A prototype black oil simulator is described. The simulator has a domain-based data structure whereby the reservoir is represented by a possibly large number of smaller reservoirs each having a complete local data structure. This design is essential for effective use of preconditioning techniques based on domain decomposition. The chapter describes a splitting technique for the solution of the nonlinear system and an effective implementation of the algorithm on massively parallel computer systems. Most communication is localized and long range communication is kept at a minimum. Results from an implementation of the method are reported for a 16384 processor MasPar MP-2. 1 Introduction This chapter describes a numerical algorithm for the simulation of flow in porous media. The problem carries substantial economic interest in the petroleum industry. It is widely used for planning purposes, for reservoir management and for prediction of the reservoir performance [3]. Another field of app..

On The Spectra Of Sums Of Orthogonal Projections With Applications To Parallel Computing

. Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product. AMSMOS: 65N30 65J10 35J20 15A18 Keywords: Orthogonal Projections, Parallel Computing, Domain Decomposition, Grid Refinement, Schwarz Alternating Method. 1. Introduction. Recently there has been a strong revival of the interest in domain decomposition algorithms for elliptic problems; cf. e.g. Glowinski et al. [12], and Chan et al. [4]. A classical algorithm of this kind is the Schwarz alte..