Convergence acceleration of Runge–Kutta schemes for solving the Navier–Stokes equations

Swanson, R. C.; Turkel, Eli; Rossow, Cord-Christian

doi:10.1016/j.jcp.2007.02.028

Cited by 77 publications

(56 citation statements)

References 31 publications

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“…This double discretization approach allows one to use high-order accuracy in the actual discretization without having to deal with the stability problems encountered in also using the high-order spatial operator for smoothing (Brandt 2001). We note that the terms containing M and L already appeared in Rossow (2007) and Swanson et al (2007) but in a different notation that should not be confused with ours. The terms with the factors C/Δt and ∂S/∂U are, however, new.…”

Section: Implicit Smoothing Stagesmentioning

confidence: 80%

“…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%

“…5.3.3, are not used, then the five-stage scheme is clearly superior to the three and two-stage schemes in terms of smoothing. If, on the other hand, implicit stages are employed, the smoothing performance of the simpler schemes is on par with the five-stage scheme (see Swanson et al 2007), which then becomes unattractive due to its larger cost. As they stand, the multistage schemes given by Eqs.…”

Section: Explicit Multistage Smoothing Schemementioning

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%

“…Rossow 2007, for details). The factor was introduced by Swanson et al (2007) as a weighting factor for the degree of implicitness of the pseudo-time discretization. It serves to optimize the smoothing properties of the scheme, and its optimal value is typically determined by a local Fourier analysis.…”

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: Implicit Smoothing Stagesmentioning

confidence: 80%

Section: A47 Page 2 Of 30mentioning

confidence: 99%

Section: Explicit Multistage Smoothing Schemementioning

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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Solution Methods and Acceleration Techniques in Computational Fluid Dynamics ( CFD )

Dwight

Lucas

Bijl

2010

Encyclopedia of Aerospace Engineering

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Methods for solving the discretized equations of fluid motion are discussed, with particular reference to efficient methods for solving compressible flow problems to a steady state. A spectrum of algorithms of widely varying efficiency and complexity are considered, from explicit single-stage methods, through Runge-Kutta, implicit methods, defect correction, to multigrid. They are classified with respect to cost per iteration, convergence rate and memory requirements, and compared for a concrete high-Reynolds number test-case. Time-resolved calculations are then considered using the dual-time framework.

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Aerodynamics

Witherden

Jameson

2017

Encyclopedia of Computational Mechanics Second Edition

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Computational methods are now pervasive in the science of aerodynamics. Because previously existing numerical methods proved inadequate for fluid flow simulations, the emergence of computational fluid dynamics (CFD) as a distinct discipline has sparked the development of an entirely new class of algorithms, and a supporting body of theory, which are the main theme of this chapter. After a review of mathematical models of fluid flow, methods for solving the transonic potential flow equation (of mixed type) are examined. The central part of the chapter discusses the formulation and implementation of shock‐capturing schemes for the Euler and Navier–Stokes equations. The chapter next discusses the merits of the contrasting approaches of finite difference, finite volume, and finite element methods for the treatment of flows in complex geometric domains, together with the treatment of boundary conditions. A detailed presentation of explicit and implicit time‐stepping schemes for both steady and unsteady flows precedes a discussion of preconditioning and multigrid procedures for accelerating the convergence of steady‐state calculations. High‐order methods suitable for conducting unsteady simulations on unstructured grids are also presented. In order to realize the potential benefits of CFD, it is essential to move beyond simulation to aerodynamic (and ultimately multidisciplinary) optimization. The chapter concludes with a discussion of aerodynamic shape optimization via control theory.

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Convergence acceleration of Runge–Kutta schemes for solving the Navier–Stokes equations

Cited by 77 publications

References 31 publications

On multigrid solution of the implicit equations of hydrodynamics

On multigrid solution of the implicit equations of hydrodynamics

Solution Methods and Acceleration Techniques in Computational Fluid Dynamics ( CFD )

Aerodynamics

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