Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow

Bijl, H.; Carpenter, Mark H.; Vatsa, Veer N.; Kennedy, Christopher

doi:10.1006/jcph.2002.7059

Cited by 198 publications

(147 citation statements)

References 14 publications

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“…Among the latter are the less-known implicit Runge-Kutta (RK) schemes. Good discussions of the advantages and disadvantages of BDF versus implicit RK schemes for the solution of fluid dynamics problems are given in Bijl et al (2002) and Jothiprasad et al (2003).…”

Section: Temporal Discretizationmentioning

confidence: 99%

“…These methods were evaluated for the solution of the Navier-Stokes equations by Bijl et al (2002) and Jothiprasad et al (2003). For the numerical experiments presented in this paper, we make use of only the ESDIRK23 and ESDIRK46 schemes.…”

Section: Singly Diagonally Implicit Runge-kutta Schemes With An Explimentioning

confidence: 99%

“…Moreover, we replace the ESDIRK23 temporal discretization by the higher-order ESDIRK46 scheme given by Kennedy & Carpenter (2001). The coefficients of the latter scheme can also be found in Bijl et al (2002). The high Courant numbers, that we consider here, preclude the use of the ESDIRK23 scheme.…”

Section: Vortex Shedding Behind a Cylindermentioning

confidence: 99%

“…This is a standard problem for Navier-Stokes codes, for which accurate comparisons to experimental results have been performed (see, e.g. Melson et al 1993;Bijl et al 2002;Jothiprasad et al 2003, and the references therein). We note that it is not our aim here to provide a physically satisfactory simulation of this problem.…”

Section: Vortex Shedding Behind a Cylindermentioning

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: Temporal Discretizationmentioning

confidence: 99%

Section: Singly Diagonally Implicit Runge-kutta Schemes With An Explimentioning

confidence: 99%

Section: Vortex Shedding Behind a Cylindermentioning

confidence: 99%

Section: Vortex Shedding Behind a Cylindermentioning

confidence: 99%

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On multigrid solution of the implicit equations of hydrodynamics

Kifonidis

Müller

2012

A&A

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“…It has already been demonstrated that for flow applications, higher order time integration schemes are computationally more efficient than popular lower order schemes, even for engineering levels of accuracy [2]. This drives the idea to use higher order time integration schemes for fluid-structure interaction simulations as well.…”

Section: Introductionmentioning

confidence: 99%

Implicit and Explicit Higher Order Time Integration Schemes for Fluid-Structure Interaction Computations

Zuijlen

Bijl

2004

Computational Science - ICCS 2004

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Abstract. In this paper higher order time integration schemes are applied to fluid-structure interaction (FSI) simulations. For a given accuracy, we investigate the efficiency of higher order time integration schemes compared to lower order methods. In the partitioned FSI simulations on a one-dimensional piston problem, a mixed implicit/explicit (IMEX) time integration scheme is employed: the implicit scheme is used to integrate the fluid and structural dynamics, whereas an explicit Runge-Kutta scheme integrates the coupling terms. The resulting IMEX scheme retains the order of the implicit and explicit schemes. In the IMEX scheme considered, the implicit scheme consists of an explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) scheme, which is a multistage, L-stable scheme.

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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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Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow

Cited by 198 publications

References 14 publications

On multigrid solution of the implicit equations of hydrodynamics

On multigrid solution of the implicit equations of hydrodynamics

Implicit and Explicit Higher Order Time Integration Schemes for Fluid-Structure Interaction Computations

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

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