A new multigrid algorithm for non-linear equations in conjunction with time-marching procedures
TL;DR: An algorithm to apply the multigrid technique to the equations linearized in time is developed exploring this possibility and is implemented for two-dimensional incompressible and compressible flows coupled with explicit time marching procedures.
Abstract: Full approximate storage (FAS) multigrid algorithm is the most commonly used multigrid algorithm for non-linear equations. The algorithm initially developed for steady-state equations was later extended to obtain steady-state solutions employing unsteady equations. In extending the FAS algorithm for the steady-state non-linear equations to unsteady non-linear equations, the FAS algorithm does not to take into account that the governing equations are typically linearized in time before they are solved. Thus, there is a scope to develop a new multigrid algorithm to apply the multigrid technique to the equations linearized in time. In the present work, such an algorithm is developed exploring this possibility and is implemented for two-dimensional incompressible and compressible flows coupled with explicit time marching procedures. The results of the new algorithm compare favourably with those of the FAS multigrid method and single grid.
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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.
Abstract: 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. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10.000 and meshes consisting of as many as 257 x 257 points. For Re = 1000, the (129 x 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional gridclustering coordinate transformations. The past decade has witnessed a great deal of progress in the area of computational fluid dynamics. Developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems. For some of these applications, the use of simple iterative techniques to solve the Navier-Stokes equations leads to a rather slow convergence rate for the solutions. The solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries. The rate of convergence is also generally strongly dependent on such problem parameters as the Reynolds number, the mesh size, and the total number of computational points. This has led several researchers to examine carefully the recently emerging multigrid (MG) technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously.
3,728 citations
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01 Jan 1984
TL;DR: In this paper, a reference record was created on 2005-11-18, modified on 2016-08-08 and used for CFD-based transfert de chaleur.
Abstract: Keywords: CFD ; numerique ; transfert de chaleur ; ecoulement Reference Record created on 2005-11-18, modified on 2016-08-08
3,596 citations
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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
2,923 citations
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01 Apr 1982
TL;DR: In this article, the authors present a method for numerique numeriques for programmation with differences between finies and viscosite reference records. But the method is not presented in detail.
Abstract: Keywords: methodes : numeriques ; programmation ; differences : finies ; methode : integrale ; ecoulement : incompressible ; ecoulement : compressible ; ecoulement : visqueux ; mecanique des : fluides ; viscosite Reference Record created on 2005-11-18, modified on 2016-08-08
1,184 citations
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01 Jan 1985
TL;DR: In this article, the authors present a method for numerique numeriques for programmation with differences between finies and viscosite reference records. But the method is not presented in detail.
Abstract: Keywords: methodes : numeriques ; programmation ; differences : finies ; methode : integrale ; ecoulement : incompressible ; ecoulement : compressible ; ecoulement : visqueux ; mecanique des : fluides ; viscosite Reference Record created on 2005-11-18, modified on 2016-08-08
1,182 citations
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