Schreiber, H., Keller, H.B.: Driven Cavity flows by efficient numerical techniques, J. Comp. Phys. 49 (1983), S. 310-333  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem has been previously analyzed for the uni processor case [1] Here, we consider the impact of non uniform memory access times imposed by parallel distributed memory architectures. Examples of architecture and discretization dependent computational cost analyses have also been presented in [2, 3, 4, 5]. Keller and Schreiber [2] study the costs for parameter continuation of steady state Navier Stokes calculations on vector supercomputers with a cost functional that incorporates both CPU and memory charges, and with the principal independent parameter being the frequency of full Newton vs. chord ....

....for the uni processor case [1] Here, we consider the impact of non uniform memory access times imposed by parallel distributed memory architectures. Examples of architecture and discretization dependent computational cost analyses have also been presented in [2, 3, 4, 5] Keller and Schreiber [2] study the costs for parameter continuation of steady state Navier Stokes calculations on vector supercomputers with a cost functional that incorporates both CPU and memory charges, and with the principal independent parameter being the frequency of full Newton vs. chord continuation steps. Chan ....

Keller, H.B. and Schreiber, R. 1983. "Driven cavity flows by efficient numerical techniques." J. Comp. Phys. 49(2):310-333.

....= 50 50 100 100 element mesh to the element mesh obtained from one level of refinement. Statistics for a sequenced run are given as sums, where each summand applies respectively to a mesh of the sequence. Example 1. Lid Driven Cavity In this standard benchmark problem [7, 11], the momentum transport and total mass conservation equations defined in Table 1 are solved on a unit square to simulate confined flow driven by a moving boundary ( on the upper wall. No slip boundary conditions are applied on all other walls. Other parameters are chosen so that the Reynolds ....

R. Schreiber and H.B. Keller. "Driven cavity flows by efficient numerical techniques." J. Comput. Phys., 49 (1983) 310-333.

....permit 2 approximation of the Jacobian matrix) over a sequence of Newton iterations, while still converging quadratically. This theory was revisited to provide inexpensive, constructive formulae for the sequence of inexact tolerances by Eisenstat Walker [25] Smooke [72] and Schreiber Keller [69] devised Newton chord methods with models for cost effective frequency of Jacobian reevaluation. The use of various approximate Newton methods in CFD emerged independently in various regimes. Vanka [82] implemented Newton solvers in primitive variable Navier Stokes problems. Venkatakrishnan [83] ....

R. Schreiber and H. B. Keller. Driven cavity flows by efficient numerical techniques. J. Computational Physics, 49:310--333, 1983.

....though the number of iterations performed can be reduced from the local version to the global one, the total running time is certainly larger. 5.2 Driven cavity flows A different set of experiments was performed using a classical problem in fluid mechanics, the cavity problem. See, for example, [52]. The problem is to find : Omega IR, Omega = 0; 1] Theta [0; 1] such that Delta 2 Re[ x 1 x 2 Delta Gamma x 2 x 1 Delta ] 0 in Omega with the boundary conditions = 0 on Omega n (x 1 ; 0) n (0; x 2 ) n (1; x 2 ) 0; ....

.... Theta [0; 1] such that Delta 2 Re[ x 1 x 2 Delta Gamma x 2 x 1 Delta ] 0 in Omega with the boundary conditions = 0 on Omega n (x 1 ; 0) n (0; x 2 ) n (1; x 2 ) 0; n (x 1 ; 1) 1: We used the discretization scheme of [52] with a 63 Theta 63 grid. The nonlinearity of the problem increases as Re, the Reynolds number, increases. We solved the problem for 45 equally spaced Reynolds numbers, from Re = 0 to Re = 11000. We used the solution obtained for each Reynolds number Re as initial point for the problem defined by ....

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Schreiber R. and Keller, H.B. (1983). Driven cavity flows by efficient numerical techniques, Journal of Computational Physics 49, 310-333. 31

....coarse grid used, which is refined four times (4;801 vertices, 14;209 unknowns, h min = 1 250 , h max = 1 48 ) resp. five times, resulting in 18;817 vertices, 56;065 unknowns, h min = 1 500 , h max = 1 96 . We chose Re = 2;000, since for this Reynolds number range (see, for instance, 3] [15]) a new secondary eddy appears at the upper left edge. In Table 1 the values and the coordinates are given for the main vortex and the secondary vortex in the lower right corner, and are compared to the values from Vanka (V) 23] and Zhang (Z) 24] both on rectangular grids) We show the ....

Schreiber, R., Keller, H.B.: Driven cavity flows by efficient numerical techniques, J. of Comp. Phys., 49, 310--333 (1983)

....F (x c ) the subsequent chord iteration, and s = GammaA Gamma1 F (x c ) the chord step. We begin with the simple observation that ks e c k ke c k = ke k ke c k : 5) Hence the relative accuracy of approximating ke c k by ksk is roughly equal to the q factor. We can conclude, as in [14], that termination on small steps is an effective approach only if the convergence is fast. From (5) one can show that if A is nonsingular and x c is near enough to x then ke k ke c k (A) kEk kAk flke c k 2kAk : 6) Therefore jksk Gamma ke c kj ke c k ks e c k ke c k = ....

R. Schriber and H. B. Keller, Driven cavity flows by efficient numerical techniques, J. Comp. Phys., 49 (1983), pp. 310--333.

....and s = GammaA Gamma1 F (x c ) the chord step. We begin with the simple observation that s = x Gamma x c = e Gamma e c , and so ks e c k ke c k = ke k ke c k : 2.4) Hence the relative accuracy of approximating ke c k by ksk is roughly equal to the q factor. We can conclude, as in [21], that termination on small steps is an effective approach only if the convergence is fast. Now if A is nonsingular, and x c is near enough to x , then e = e c Gamma A Gamma1 Z 1 0 F 0 (x te c )e c dt = A Gamma1 Z 1 0 (A Gamma F 0 (x te c ) e c dt = A Gamma1 (A ....

R. Schriber and H. B. Keller, Driven cavity flows by efficient numerical techniques, J. Comp. Phys., 49 (1983), pp. 310--333.

....1 is subtle. If one is solving the equation for the Newton step with a direct method, then evaluation and factorization of the Jacobian matrix is not done at every timestep. This is a common feature of many ODE and DAE codes, 30, 26, 27, 3] Jacobian updating is an issue in continuation methods [31, 28], and implementations of the chord and Shamanskii [29] methods for general nonlinear equations [2, 17, 25] When the Jacobian is slowly varying as a function of time or the continuation parameter, sporadic updating of the Jacobian leads to significant performance gains. One must decide when to ....

R. SCHRIBER AND H. B. KELLER, Driven cavity flows by efficient numerical techniques, J. Comp. Phys., 49 (1983), pp. 310--333.

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Schreiber, H., Keller, H.B.: Driven Cavity flows by efficient numerical techniques, J. Comp. Phys. 49 (1983), S. 310-333

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R. Schriber and H. B. Keller, Driven cavity flows by efficient numerical techniques, J. Comp. Phys., 49 (1983), pp. 310--333.

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Schreiber, R. and Keller, H. B., "Driven CavityFlows by Efficient Numerical Techniques," Journal of Computational Physics,Vol. 49, 1983, pp. 310--333.

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H. Schreiber and H.B. Keller. Driven cavity flows by efficient numerical techniques. Journal of Computational Physics, 49:310--333, 1983.

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