A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput., 15 (1994), pp. 681--703.  Home/Search   Document Not in Database   Summary   Related Articles   Check

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A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput., 15 (1994), pp. 681--703.

....residual norm improves only slightly. This provides direct feedback limiting the increase of the timestep (and possibly decreasing it) which maintains or improves the linear conditioning of the next step, rather than letting the conditioning deteriorate with increasing pseudo timestep. See [26] for a polyalgorithmic method that exploits the effect of the pseudo timestep on the linear conditioning. We have experimented a good deal with these strategies, and we find that a hybrid approach is the most cost effective in large transonic flow problems. Such an approach involves an initially ....

A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke. Towards polyalgorithmic linear system solvers for nonlinear elliptic systems. SIAM J. Scientific Computing, 15:681--703, 1994.

....residual norm improves only slightly. This provides direct feedback limiting the increase of the timestep (and possibly decreasing it) which maintains or improves the linear conditioning of the next step, rather than letting the conditioning deteriorate with increasing pseudo timestep. See [26] for a polyalgorithmic method that exploits the e#ect of the pseudo timestep on the linear conditioning. We have experimented a good deal with these strategies, and we find that a hybrid approach is the most cost e#ective in large transonic flow problems. Such an approach involves an initially ....

A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke. Towards polyalgorithmic linear system solvers for nonlinear elliptic systems. SIAM J. Scientific Computing, 15:681--703, 1994.

....elliptic BVPs. This field is fairly wide open for other types of numerical analyses. Fidelity based multi model methods have been used in ad hoc ways in numerous commercially important engineering codes, e.g. Boeing TRANAIR [5] Polyalgorithmic solvers have been demonstrated in principle e.g. [3], but rarely in the hostile environment of high performance multiprocessing. These advanced adaptive approaches demand sophisticated software approaches, such as object oriented programming. Management of hierarchical levels of synchronization (within a region and between regions) is also ....

A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke. Towards polyalgorithmic linear system solvers for nonlinear elliptic systems. SIAM J. Sci. Comput., 15:681--703, 1994.

....of highly nonlinear problem, linear methods making use of varying amounts of global information may be employed along the route as Deltat advances from 0 toward 1. This issue was explored theoretically for the heat equation and experimentally for a two dimensional laminar reacting flow in [14], where it was concluded that a polyalgorithmic linear solver simple iterations like point Gauss Seidel at the outset and accelerated methods like preconditioned GMRES in the endgame will often lead to the best overall complexity in a difficult steady nonlinear problem. An interesting way ....

A. Ern, V. Giovangigli, D. E. Keyes and M. D. Smooke (1994): Towards Polyalgorithmic Linear System Solvers for Nonlinear Elliptic Problems, SIAM J. Sci. Comp. 15 681--703.

....to the residual reduction. # n = # 0 #F (x 0 )# #F (x n )#. 1.5) Relation (1.5) implies that, for n # 1, # n = # n 1 #F (x n 1 )# #F (x n )# . In some work [16] # n is kept below a large, finite bound # max . Sometimes # n is set to # (called switchover to steady state form in [13]) when the computed value of # n exceeds # max . In view of these practices, we will allow for somewhat more generality in the formulation of the sequence # n . We will assume that # 0 is given and that # n = # # n 1 #F (x n 1 )# #F (x n )# (1.6) for n # 1. The choice in [24] ....

A. ERN, V. GIOVANGIGLI, D. E. KEYES, AND M. D. SMOOKE, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput., 15 (1994), pp. 681-- 703.

....there will usually be an asymptotic regime in which the power of Newton s method is desirable if the storage overhead is not too great. To connect the opening iterations to the asymptotic regime, polyalgorithmic linear solvers for the Newton corrections were shown to be desirable in, for instance, [8]. Regarding (2) we refer to [19] for recent developments. The last three considerations are the most important with respect to parallel CFD. For a variety of reasons, industrial CFD groups are inclining towards the distributed network computing environment characterized by coarse to medium ....

A. Ern, V. Giovangigli, D. E. Keyes and M. D. Smooke, Towards Polyalgorithmic Linear System Solvers for Nonlinear Elliptic Problems, SIAM J. Sci. Comp. 15(1994), to appear.

.... mesh schemes have drawbacks when employing complex geometric configurations requiring non orthogonal curvilinear coordinates (Armfield, 1991, Sotiropoulos and Abdallah, 1992) The vorticity velocity formulation is a relatively new formulation in the context of chemically reacting flows (Ern, 1994). Although traditionally used for incompressible fluid flow computations (see Gatski, 1991, for a review) this formulation has been recently extended to two and three dimensional compressible flows (Ern and Smooke, 1993, Guevremont et al. 1993, Ern, 1994) It eliminates the pressure while ....

.... the context of chemically reacting flows (Ern, 1994) Although traditionally used for incompressible fluid flow computations (see Gatski, 1991, for a review) this formulation has been recently extended to two and three dimensional compressible flows (Ern and Smooke, 1993, Guevremont et al. 1993, Ern, 1994). It eliminates the pressure while replacing the first order continuity equation with additional second order equations. Unlike stream function vorticity, vorticity velocity is easily extendible to three dimensions and allows more accurate formulation of boundary conditions in a numerically ....

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Ern, A., Giovangigli, V., Keyes, D. E., and Smooke, M. D. 1994. Towards polyalgorithmic linear system solvers for nonlinear elliptic problems. SIAM J. Sci. Comput. 15:681--703.

.... this paper, we consider instead two generalizations of the conjugate gradient method to nonsymmetric systems: the Bi CGSTAB algorithm [25] and the restarted version of GMRES [20] The effectiveness of both iterative solvers has been illustrated clearly for chemically reacting flow problems in [6] [8]. Bi CGSTAB has lower memory requirements but, because of the lack of an optimality property of the residual, oscillatory behavior may be encountered, especially on fine grids. On the other hand, GMRES minimizes the residual norm on a Krylov space but work and storage grow linearly per iteration ....

....on a Krylov space but work and storage grow linearly per iteration so that, in practice, a restarted version is used. Both iterative solvers are preconditioned with a Gauss Seidel (GS) left preconditioner. The effectiveness of this preconditioner has been illustrated on a flame sheet problem in [8]. The Jacobian matrices J ( Phi n ) are nine block operators, where the blocks are dense square matrices of size equal to the number of components in the calculation. Therefore, a preconditioned matrix vector multiply consists of a lower triangular system solve combined with a block tridiagonal ....

[Article contains additional citation context not shown here]

A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput., 15 (1994), pp. 681--703.

....reduction. ffi n = ffi 0 kF (x 0 )k=kF (x n )k: 1.5) Relation (1.5) implies that, for n 1, ffi n = ffi n Gamma1 kF (x n Gamma1 )k kF (x n )k : In some work [16] ffi n is kept below a large, finite bound ffi max . Sometimes ffi n is set to 1 (called switchover to steady state form in [13]) when the computed value of ffi n exceeds ffi max . In view of these practices, we will allow for somewhat more generality in the formulation of the sequence fffi n g. We will assume that ffi 0 is given and that ffi n = OE ffi n Gamma1 kF (x n Gamma1 )k kF (x n )k : 1.6) for n 1. The ....

A. ERN, V. GIOVANGIGLI, D. E. KEYES, AND M. D. SMOOKE, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput., 15 (1994), pp. 681--703.

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