W.A. Mulder, 1989. A new multigrid approach to convection problems. J. Comput. Phys., 83: 303--323. 33  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on problems that have anisotropic discrete operators. Several methods to deal with anisotropic operators have been studied in the multigrid literature. One popular approach is semicoarsening where the multigrid coarsening process is not applied uniformly to all of the coordinate directions [13, 14, 16, 19]. By selectively not coarsening the grid in certain directions, the anisotropy can be reduced on the coarser grids. This process makes the smoothing problem easier because only part of the high frequency error (namely, the error components oscillating in the directions of coarsening) should be ....

W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys., 83 (1989), pp. 303-- 323.

....for managing the grid calculation. E, evaluate the change in the flow for one step; T,transfer the data without updating the solution. tend to slow down as the calculation proceeds to a low asymptotic rate. This has motivated the introduction of semi coarsening and directional coarsening methods [132, 133, 3, 4, 5, 148, 149]. The multigrid method can be applied on unstructured meshes by interpolating between a sequence of separately generated meshes with progressively increasing cell sizes [91, 126, 127, 146] It is not easy to generate very coarse meshes for complex configurations. An alternative approach, which ....

W. Mulder, A new multigrid approach to convection problems, J. Comp. Phys., 83 (1989), pp. 303-- 323.

....algorithm suitable for SIMD machines on the CM 2. His method requires half the amount of relaxation computation as that required in the full coarsening multigrid case, while it requires line or plane relaxation. An alternative way of achieving robustness is to use multiple coarse grids. Mulder [38] and Naik [39] proposed a multiple semicoarsed grid (MSG) algorithm and Overman [44] implemented this algorithm on distributed memory machines. This method has ample parallelism and relative robustness. However, the MSG algorithm needs a larger amount of computation than the ordinary multigrid ....

Mulder, W. A., "A new multigrid approach to convection problems," J. Comp. Phys., vol. 83, pp. 303--323, 1989.

....problem. To improve robustness several modi cations have been proposed in the literature, such as robust smoothers (smoothers which try to follow the ow direction) matrix dependent prolongations and restrictions and semicoarsening techniques. For an explanation of these methods we refer to [9, 27, 3, 7, 13, 14, 16, 17, 21, 30]. These modi cations are based on heuristic arguments and empirical studies. Related to the theoretical analysis of multigrid applied to convection di usion problems we note the following. In the literature one nds convergence analyses of multigrid methods for nonsymmetric elliptic boundary value ....

Mulder, W. (1989): A new multigrid approach to convection problems. J. Comput. Phys. 83, 303-323

....every smoothing step on the artificial boundaries as in [11] but if there is strong coupling between the subsystems convergence is still slow. As described in [13] non standard multigrid techniques have been developed to overcome some of these difficulties, such as multiple semi coarsened grids ([12]) which generates more coarse grids than standard multigrid. This is achieved, for example, by only coarsening in one direction. In a SIMD parallel environment the implementation problems are entirely different, but the main advantage is that within each step the same process is being applied to ....

W.A. Mulder, A New Multigrid Approach to Convection Problems, J. Comp. Phys., 83, (1989), 303--323.

....is only performed along the dominant direction. The meshsize along the other direction is not coarsened. In case the dominant direction is not known a priori, semicoarsening is to be performed in both directions. This will generate two coarse grids, each of which coarsens in one direction [15, 16]. To avoid using alternating line relaxation or generating double coarse grids and to facilitate parallel processing, a more thoughtful strategy is to combine line relaxation in one direction with semicoarsening in another direction to achieve robustness [20, 4] For the particular problem ....

....with different color can be updated independently. This implementation embodies inherent parallelism to Gauss Seidel relaxation [1] Experimental results also show that four color Gauss Seidel relaxation has better smoothing effect than lexicographical Gauss Seidel relaxation [10] Mulder [15] argues that since point relaxation has no effect on the weak direction (here the y direction) the residual (error) components along the y direction may not be smoothed. If the initial residual contains high frequency components in the y direction, these high frequency components will remain. ....

W. A. Mulder. A new multigrid approach to convection problems. J. Comput. Phys., 83(2):303--323, 1989.

.... of multigrid convergence in highly stretched boundary layer cells, Allmaras proposed the use of an implicit ADI preconditioner for full coarsening multigrid and preconditioners based on point implicit block Jacobi and semi implicit lineJacobi for the semi coarsening multigrid algorithm of Mulder [1, 2, 14, 15]. These suggestions are motivated by the realization that for efficient multigrid performance, the relaxation scheme on each mesh must damp all modes which cannot be resolved without aliasing on the next coarser mesh in the cycle. Preconditioning is intended to cluster the residual eigenvalues of ....

....modes that are high frequency in one mesh direction and low frequency in the other. Allmaras recommends an implicit preconditioner because explicit methods are notoriously poor at damping modes with a low frequency component [2] Alternatively, the semi coarsening algorithm proposed by Mulder [14, 15] coarsens separately in each mesh direction and therefore reduces the region of Fourier space for which the relaxation scheme on each mesh must successfully damp modes for the algorithm to function efficiently. To obtain an O(N) method for a 3D mesh with N points, Mulder defined a restriction and ....

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W.A. Mulder. A new multigrid approach to convection problems. J. Comp. Phys., 83:303-- 323, 1989.

....coecients that behave like O(r) and O(r 1 ) close to the axis, i.e. for r 0. Ellipticity degrades and so does multigrid convergence. However, the awkward behavior of the coecients is aligned with the radial coordinate direction r. This special type of anisotropy suggests a standard remedy [12, 17, 24, 30, 33, 42]: Use a discretization based on a tensor product grid aligned with the coordinate axes Employ collective r line relaxation as a smoother. Rely on semicoarsening in z direction to de ne the coarse grids. Throughout this presentation the multigrid algorithms are devised according to these ....

W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys., 83 (1989), pp. 303-323.

....functions of all grids that result from semi refinement steps with respect to both coordinate directions. Furthermore, the full and the sparse grid case [14] is considered. For the level oriented methods, we obtain multigrid algorithms similar to that of Naik and van Rosendale [12] and Mulder [11]. For the point and domain oriented approach, analogous point block Gauss Seidel methods as well as BPX like preconditioners are derived. ....

W. Mulder, A new multigrid approach to convection problems, CAM Report 88-04, (1988).

....Subsequent application this process in all coordinate directions makes a partially ordered set of grids rather than a sequentially ordered one. All grids that can be seen as the coarsenings of one particular grid are called a full grid of grids. Semi coarsening is earlier described for d = 2 in [9, 10] and for d = 3 in [1, 2, 3, 6, 7, 8] It is an additional disadvantage of regular refinement in each direction, especially for higher dimensions, that the number of the degrees of freedom increases very fast when more levels of refinement are introduced. In the battle against the increasing number ....

W.A. Mulder. A New Multigrid Approach to Convection Problems. Journal of Computational Physics, 83:303--323, 1989.

....of time step values defining multi stage schemes with superior smoothing properties, for discretizations of the full Euler or Navier Stokes spatial operator, provided these incorporate local preconditioning. With the use of semi coarsening rather than fullcoarsening in the multigrid process [40], the high frequency domain over which the multi stage scheme must be a good damper of errors shrinks to high high combinations only. This makes the multi stage schemes largely independent of flow conditions such as Mach number and flow angle. The resulting multi stage schemes are not only ....

.... itself already accelerates the convergence to a steady solution (this benefit persists in multigrid relaxation) and the highfrequency damping provides robustness [34, 35, 36] This is a substantial improvement over the standard block Jacobi preconditioning recommended for multigrid use by Mulder [40] and Allmaras [2] which yields equally good high frequency damping but no equalization of characteristic speeds. The spatial discretization techniques that we have selected in order to achieve high resolution, efficiency and robustness are upwind based discretizations based on 5 ....

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W. Mulder, "A new multigrid approach to convection problems," Journal of Computational Physics, vol. 83, 1989. 193

....of grids for MSG. The semi coarsening technique is well known and used especially in the structured mesh community. For complex geometries, however, multiple directions within the mesh require semi coarsening. A process named Multiple Semicoarsened Grid (MSG) Algorithm was introduced by Mulder [9]. This technique relies on the generation of numerous grids that are semi coarsened (SC) from the finer grid in all possible directions as depicted in Fig.3. This ensures proper dissipation of the signal. A multigrid scheme is then implemented using all the grids which is complex and costly, ....

W. Mulder. A New Multigrid Approach to Convection Problems. Journal of Computational Physics, 83:303--329, 1989.

....grids leads to a multiple, intertwined, coarse grid correction. Again, at the expense of a larger complexity on the coarse grid, one obtains an effective CGC which clears the way for a simple smoothing operator. In 2D, Mulder proposed to perform semi coarsening in two directions simultaneously [15, 16]. A fine grid is coarsened in the x and y direction respectively. Vice versa, each coarse grid is linked to two finer grids, refined in the x and y direction respectively. Similar information from different coarse grids is combined in order to limit the complexity to O(N) where N is the number ....

....is not uniquely defined any more because Rn;n e1 dn e1 6= Rn;n e2 dn e2 and some weighted averaging of these restricted residuals would have to be introduced for the computation of dn . It is not clear in advance that equal weighting: dn : 1 2 Rn;n e1 dn e1 1 2 Rn;n e2 dn e2 (see e.g. [15, 17]) would be the appropriate choice in all cases possible. e.g. consider the particular situation that kdn e1 k AE kdn e2 k due to some odd behaviour of the pre relaxation method. Equal weighting for the restricted residuals in stage A then results in a too large correction for un e2 and a too ....

W.A. Mulder, A new multigrid approach to convection problems, J. Comput. Phys. 83 (1989) 303--323.

....propagative disparities in the limit of vanishing Mach number. In certain cases, preconditioning methods can also be used to alleviate the problem of directional decoupling [22, 14, 26] Another method for countering directional decoupling is the use of directional coarsening multigrid algorithms [27]. The interaction between the preconditioner and the multigrid coarsening algorithm is critical, making it imperative that the two components of the scheme are considered simultaneously when attempting to design efficient preconditioned multigrid methods. Allmaras provided a systematic ....

....when attempting to design efficient preconditioned multigrid methods. Allmaras provided a systematic examination of the damping requirements for relaxation methods used in conjunction with both the traditional full coarsened multigrid and for the semi coarsening multigrid algorithm of Mulder [26, 27]. Using full coarsened multigrid in two dimensions, only modes which are low frequency in both mesh directions can be resolved on the coarser grids, so that the relaxation scheme must eliminate all high frequency modes, and also those modes that are high frequency in one mesh direction and low ....

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W.A. Mulder. A new multigrid approach to convection problems. J. Comput. Phys., 83:303--323, 1989.

....multi stage scheme can provide rapid damping and propagation. In certain cases, preconditioning methods can also be used to alleviate the problem of directional decoupling [17, 14, 9] Another method for countering directional decoupling is the use of directional coarsening multigrid algorithms [18]. The interaction between the preconditioner and the multigrid coarsening algorithm is critical, making it imperative that the two components of the scheme are considered simultaneously when attempting to design efficient preconditioned multigrid methods. Allmaras provided a systematic ....

....when attempting to design efficient preconditioned multigrid methods. Allmaras provided a systematic examination of the damping requirements for relaxation methods used in conjunction with both the traditional full coarsened multigrid and for the semi coarsening multigrid algorithm of Mulder [9, 18]. Using full coarsened multigrid in two dimensions, only modes which are low frequency in both mesh directions can be resolved on the coarser grids, so that the relaxation scheme must damp all high frequency modes, and also those modes that are high frequency in one mesh direction and low ....

[Article contains additional citation context not shown here]

W.A. Mulder. A new multigrid approach to convection problems. J. Comp. Phys., 83:303--323, 1989.

....the effectiveness of multigrid procedures which can not eliminate all the error modes which can exist in the solution. To overcome this drawback, different methods have been proposed. One is a semi coarsening multigrid strategy, in which the mesh is not coarsened in every direction simultaneously [10], while others are based on the use of a preconditioner [1] which has the effect of moving the eigenvalues away from the origin of the Fourier complex plan providing, within an optimised Runge Kutta update, a very good damping of the high frequency error modes. Recently, Pierce and Giles [12] ....

W.A. Mulder. A new multigrid approach to convection problems. Journal of Computational Physics, 83:303--323, 1989.

....convection diffusion equation. The classical multilevel V cycle applied to equation (2) diverges if ffl tends to 0 because the restricted coarse grid equations of equation (2) are not stable. There are two ways to avoid this problem. One is to improve the coarse grid correction (see for example [10], 13] or [16] and another is to improve the smoother (see for example [7] 8] or [19] This shows that there are several multilevel algorithms which are fast independent of the size of the convection term and the number of unknowns. But there is no theory which can prove a robust ....

W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys., 83 (1989), pp. 303--323.

....application of this process in all coordinate directions makes a partially ordered set of grids rather than a sequentially ordered one. All grids that can be seen as the (semi ) coarsenings of one particular grid are called a full family of grids. Semi coarsening is earlier described for d = 2 in [16,17] and for d = 3 in 1070 5325 97 010001 21 15.50 Received 23 September 1997 c fl1997 by John Wiley Sons, Ltd. Revised March 6, 2 J. Noordmans and P.W. Hemker [1,2,3,12,11,13,14,15] It is an additional disadvantage of regular refinement in each direction, especially for higher dimensions, ....

W.A. Mulder. A New Multigrid Approach to Convection Problems. Journal of Computational Physics, 83:303--323, 1989.

....already for a relatively small number of levels. It is the curse of 3D that n 3 is already large for computer resources when n is still modest. A second remedy against slow convergence is semi coarsening. Here, grids are refined by halving (or doubling) the mesh size in one direction only [14,15,17]. Now the grids do not make an ordered sequence. The family of grids is only partially ordered. In a semi coarsening algorithm there is a finest grid for which the solution is eventually found. Again, discrete problems on the family of coarser grids are solved to accelerate the solution process. ....

....Figures 1a and 1b show 3D standard coarsening and multiple semi coarsening, respectively. Though multigrid with multiple semi coarsening is expected to be most fruitful for 3D problems, as far as we know, applications have only been published for 2D. Pioneering work has been done by Mulder [14], who has introduced multiple semi coarsening to overcome the poor convergence results observed in computing nearly gridaligned flows governed by the steady, 2D Euler equations. In [17] Radespiel and Swanson continue research on Mulder s approach for the steady, 2D Euler equations. In the present ....

W. A. Mulder. A new multigrid approach to convection problems. J. Comput. Phys., 83:303--323, 1989.

....p, and a restriction operator, r, which are both based on second order interpolation. On the finest grid level there are M Theta N cells. If there are M=2 Theta N=2 cells on the second finest level etc. we have full coarsening. Another possibility is to use semicoarsening, suggested in [10], where the grid is coarsened only in one direction at a time. 12 ERIK STERNER 7 Numerical experiments Here we solve the linearized Navier Stokes equations (2.6) on the unit square, where there is a plate at y = 0 and the other boundaries are open. The iterations are terminated when the norm ....

W. A. Mulder, A new multigrid approach to convection problems, Journal of Computational Physics, 83 (1989), pp. 303--323.

....are based on second order interpolation. On the finest grid level there are M ThetaN cells. Assuming full coarsening we have M=2 ThetaN=2 cells on the second finest level, where a coarse cell is formed by amalgamating four cells from the fine level. We also consider semi coarsening, suggested in [22], where the grid is coarsened only in the body normal direction. Thereby the aspect ratio of the cells in the boundary layer is improved on the coarser levels. In the numerical experiments we solve the stationary Navier Stokes equations linearized around a Blasius profile. We start with a test ....

W. A. Mulder, A new multigrid approach to convection problems, Journal of Computational Physics, 83 (1989), pp. 303--323.

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W.A. Mulder, 1989. A new multigrid approach to convection problems. J. Comput. Phys., 83: 303--323. 33

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W. A. Mulder, A new multigrid approach to convection problems, Journal of Computational Physics, 83 (1989), pp. 303--323.

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W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys., 83 (1989), pp. 303-- 38 323.

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W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys., 83 (1989), pp. 303--323.

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