Characteristic Multigrid Method Application to Solve the Euler Equations with Unstructured and Unnested Grids

Leclercq, Marie-Pierre; Stoufflet, B.

doi:10.1006/jcph.1993.1035

Cited by 28 publications

(14 citation statements)

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“…The method, which uses full approximation MG (FAS-MG), applies successively the subiterative procedure of the basic SG scheme [14,15] on each grid, using a V(1, 0)-cycle (sawtooth cycle) [74,77]. The restriction (transfer) and prolongation (interpolation) operators are based on the characteristic MG approach of Leclercq and Stoufflet [78], which maintains the upwind character of the scheme and ensures stability of the multigrid algorithm. The method [17] has been applied to several 2-D and 3-D flows, including shock wave/turbulent boundary layer interactions (SWTBLI) [17,18,[79][80][81], flows with large separation [17,82], and multistage transonic turbomachinery flows [18], and has consistently given CPU-speed-ups in the range r CPUSUP ∈ [3,4].…”

Section: Fully Coupled Mg With Stabilization Fixes (Fcsf-mg): In Thismentioning

confidence: 99%

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Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

Gerolymos

Vallet

2008

Numerical Methods in Fluids

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SUMMARYThe present paper investigates the multigrid (MG) acceleration of compressible Reynolds-averaged Navier-Stokes computations using Reynolds-stress model 7-equation turbulence closures, as well as lowerlevel 2-equation models. The basic single-grid SG algorithm combines upwind-biased discretization with a subiterative local-dual-time-stepping time-integration procedure. MG acceleration, using characteristic MG restriction and prolongation operators, is applied on meanflow variables only (MF-MG), turbulence variables being simply injected onto coarser grids. A previously developed non-time-consistent (for steady flows) full-approximation-multigrid (s-MG) is assessed for 3-D anisotropy-driven and/or separated flows, which are dominated by the convergence of turbulence variables. Even for these difficult test cases CPU-speed-ups r CPUSUP ∈[3, 5] are obtained. Alternative, potentially time-consistent approaches (unsteady u-MG), where MG acceleration is applied at each subiteration, are also examined, using different subiterative strategies, MG cycles, and turbulence models. For 2-D shock wave/turbulent boundary layer interaction, the fastest s-MG approach, with a V(2, 0) sawtooth cycle, systematically yields CPU-speed-ups of 5± 1 2 , quasi-independent of the particular turbulence closure used.

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Section: Fully Coupled Mg With Stabilization Fixes (Fcsf-mg): In Thismentioning

confidence: 99%

“…The 195 prolongation and restriction operators are based on characteristic MG [78]. The restriction (transfer) operator from the fine to the coarse grid, for the variables, T 2h u;h , is a simple injection operator.…”

Section: Meanflow-multigridmentioning

confidence: 99%

Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

Gerolymos

Vallet

2008

Numerical Methods in Fluids

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“…However, in the case of Chan's dual algorithm not only are the triangles not nested but the vertices as well. Practical applications for solving the Euler equations [12] with nonnested meshes do not produce in general as good convergence rates as those obtained when the grids are at least node nested [16]. However and as mentioned previously, those techniques cannot be directly applied to highly stretched meshes or to anisotropic problems.…”

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confidence: 96%

Coarsening Strategies for Unstructured Multigrid Techniques with Application to Anisotropic Problems

Morano¹,

Mavriplis²,

Venkatakrishnan³

1998

SIAM J. Sci. Comput.

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Abstract. Over the years, multigrid has been demonstrated as an efficient technique for solving inviscid flow problems. However, for viscous flows, convergence rates often degrade. This is generally due to the required use of stretched meshes (i.e., the aspect ratio AR = ∆y/∆x << 1) in order to capture the boundary layer near the body. Usual techniques for generating a sequence of grids that produce proper convergence rates on isotropic meshes are not adequate for stretched meshes. This work focuses on the solution of Laplace's equation, discretized through a Galerkin finite-element formulation on unstructured stretched triangular meshes. A coarsening strategy is proposed and results are discussed.Key words. multigrid method, unstructured meshes, semicoarsening, anisotropic problems AMS subject classifications. 65M55, 65M60, 76M10PII. S1064827595287638Introduction. Multigrid methods are well known to be successful for solving elliptic problems. This is mainly due to their good damping properties which result from two very simple principles. A usual Fourier analysis demonstrates that most of the commonly used solvers effectively damp the high frequencies of a signal. A low frequency component of a given signal on a fine mesh becomes a high frequency on a coarser one, hence the idea of solving the same problem on a sequence of meshes where all frequencies can be damped equally and, if enough grids are available, only a few iterations will be required to produce a converged solution (for more details see [4]). Despite these rather simple considerations, the multigrid algorithm is complex and difficult to implement. One of the difficulties resides in the generation of the sequence of grids for unstructured meshes. The convergence properties of the multigrid method depend upon the "quality" of these grids.A sequence of meshes may be produced through two different methods. First, starting from a mesh that is not too fine but correctly represents the problem, finer meshes may be generated through refinement. A global refinement, performed through local subdivision of the triangles of the discretization, tends to preserve the geometrical features required to obtain an efficient multigrid method. However, this will clearly not be efficient in terms of computational cost, hence the local refinement technique where specific regions of the mesh are refined and then possibly adapted [20]. Although this method seems more reasonable, it increases the computational time and the complexity of the multigrid algorithm.Another method consists in coarsening an existing fine mesh, which has been created to represent accurately the different phenomena to be observed. One of the techniques available consists in removing, through a coarsening criterion, a certain

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“…The contravariant velocity between two coarse grid cell centres is used as a sensor to switch between upwind and central prolongation if supersonic or subsonic flow exists between those two points. Again Leclercq and Stoufflet 30 describe a more accurate characteristic prolongation. However, the computational effort is much larger for such a treatment and good results have been obtained with the described simple switching.…”

mentioning

confidence: 97%

“…10 A different but more complicated possibility would be a characteristic restriction operator as proposed by Leclercq and Stoufflet 30 which requires computationally expensive matrix-vector multiplications. To locate regions where the transferred residuals should be reduced, shock sensors in the (24) is replaced by…”

mentioning

confidence: 99%

Multigrid Convergence Acceleration for Turbulent Supersonic Flows

Gerlinger

Brüggemann

1997

Int. J. Numer. Meth. Fluids

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Characteristic Multigrid Method Application to Solve the Euler Equations with Unstructured and Unnested Grids

Cited by 28 publications

References 0 publications

Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

Coarsening Strategies for Unstructured Multigrid Techniques with Application to Anisotropic Problems

Multigrid Convergence Acceleration for Turbulent Supersonic Flows

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