Convergence Acceleration for Solving the Compressible Navier-Stokes Equations

Rossow, Cord-Christian

doi:10.2514/1.15636

Cited by 22 publications

(43 citation statements)

References 21 publications

(25 reference statements)

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“…Our preferred scheme among this family of multigrid methods is based on a smoother that employs multistage pseudo-time marching with implicit smoothing stages (Rossow 2006(Rossow , 2007Swanson et al 2007). This scheme is remarkably robust for the solution of steady-state problems governed by the Euler equations.…”

Section: Discussionmentioning

confidence: 99%

“…it basically consists of an explicit integrator for ordinary differential equations. The smoothing and stability properties of this basic scheme, however, are significantly enhanced by the use of implicit stages, as proposed by Rossow (2006Rossow ( , 2007 and Swanson et al (2007).…”

Section: A47 Page 2 Of 30mentioning

confidence: 99%

“…This needs to be done carefully in order not to sacrifice their simplicity and excellent parallelizability. The approach that we propose here is based on an extension of a method introduced by Rossow (2006Rossow ( , 2007, whose basic idea can be understood as follows.…”

Section: Implicit Smoothing Stagesmentioning

confidence: 99%

“…The resulting multistage-implicit schemes were, moreover, found to work very well for fluid dynamics problems. Rossow (2007) Rossow (2006Rossow ( , 2007 and Swanson et al (2007) formulated for stationary problems, to the full dual-time Eq. (58).…”

Section: Implicit Smoothing Stagesmentioning

confidence: 99%

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On multigrid solution of the implicit equations of hydrodynamics

Kifonidis

Müller

2012

A&A

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Aims. We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics. Schemes of this class are free of the Courant-Friedrichs-Lewy condition. They are intended for simulations in which widely differing wave propagation timescales are present. A preferred solver in this class is identified. Applications to some simple stiff test problems that are governed by the compressible Euler equations, are presented to evaluate the convergence behavior, and the stability properties of this solver. Algorithmic areas are determined where further work is required to make the method sufficiently efficient and robust for future application to difficult astrophysical flow problems. Methods. The basic equations are formulated and discretized on non-orthogonal, structured curvilinear meshes. Roe's approximate Riemann solver and a second-order accurate reconstruction scheme are used for spatial discretization. Implicit Runge-Kutta (ESDIRK) schemes are employed for temporal discretization. The resulting discrete equations are solved with a full-coarsening, nonlinear multigrid method. Smoothing is performed with multistage-implicit smoothers. These are applied here to the time-dependent equations by means of dual time stepping. Results. For steady-state problems, our results show that the efficiency of the present approach is comparable to the best implicit solvers for conservative discretizations of the compressible Euler equations that can be found in the literature. The use of red-black as opposed to symmetric Gauss-Seidel iteration in the multistage-smoother is found to have only a minor impact on multigrid convergence. This should enable scalable parallelization without having to seriously compromise the method's algorithmic efficiency. For time-dependent test problems, our results reveal that the multigrid convergence rate degrades with increasing Courant numbers (i.e. time step sizes). Beyond a Courant number of nine thousand, even complete multigrid breakdown is observed. Local Fourier analysis indicates that the degradation of the convergence rate is associated with the coarse-grid correction algorithm. An implicit scheme for the Euler equations that makes use of the present method was, nevertheless, able to outperform a standard explicit scheme on a time-dependent problem with a Courant number of order 1000. Conclusions. For steady-state problems, the described approach enables the construction of parallelizable, efficient, and robust implicit hydrodynamics solvers. The applicability of the method to time-dependent problems is presently restricted to cases with moderately high Courant numbers. This is due to an insufficient coarse-grid correction of the employed multigrid algorithm for large time steps. Further research will be required to help us to understand and overcome the observed multigrid convergence difficulties for time-dependent problems.

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Section: Discussionmentioning

confidence: 99%

Section: A47 Page 2 Of 30mentioning

confidence: 99%

Section: Implicit Smoothing Stagesmentioning

confidence: 99%

Section: Implicit Smoothing Stagesmentioning

confidence: 99%

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On multigrid solution of the implicit equations of hydrodynamics

Kifonidis

Müller

2012

A&A

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“…Rossow (2006); Swanson et al (2007); Langer and Li (2011);Langer (2013) etc. We have chosen the LU-SGS iterative method to solve the system.…”

Section: Implicit Time-steppingmentioning

confidence: 99%

Acceleration on stretched meshes with line-implicit LU-SGS in parallel implementation

Otero

Eliasson

2015

International Journal of Computational Fluid Dynamics

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The implicit Lower-Upper Symmetric Gauss-Seidel solver is combined with the line-implicit technique to improve convergence on the very anisotropic grids necessary for resolving the boundary layers. The computational fluid dynamics code used is Edge, a Navier-Stokes flow solver for unstructured grids based on a dual grid and edge-based formulation. Multigrid acceleration is applied with the intention to accelerate the convergence to steady state. Lower-Upper Symmetric Gauss-Seidel works in parallel and gives better linear scaling with respect to the number of processors, than the explicit scheme. The ordering techniques investigated have shown that node numbering does influence the convergence and that the orderings from Delaunay and advancing front generation were among the best tested. 2D ReynoldsAveraged Navier-Stokes computations have clearly shown the strong efficiency of our novel approach line-implicit Lower-Upper Symmetric Gauss-Seidel which is four times faster than implicit Lower-Upper Symmetric Gauss-Seidel and line-implicit Runge-Kutta. Implicit Lower-Upper Symmetric Gauss-Seidel for Euler and line-implicit Lower-Upper Symmetric Gauss-Seidel for Reynolds-Averaged Navier-Stokes are at least twice faster than explicit and line-implicit Runge-Kutta, respectively, for 2D and 3D cases. For 3D Reynolds-Averaged Navier-Stokes, multigrid did not accelerate the convergence and therefore may not be needed.

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A Mach‐uniform preconditioner for incompressible and subsonic flows

Bas

Tuncer

Kaẏnak

2013

Numerical Methods in Fluids

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SUMMARY In this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.

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Convergence Acceleration for Solving the Compressible Navier-Stokes Equations

Cited by 22 publications

References 21 publications

On multigrid solution of the implicit equations of hydrodynamics

On multigrid solution of the implicit equations of hydrodynamics

Acceleration on stretched meshes with line-implicit LU-SGS in parallel implementation

A Mach‐uniform preconditioner for incompressible and subsonic flows

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