Accelerated convergence of Jameson's finite-volume Euler scheme using van der Houwen integrators

Pike, J.; Roe, Philip L.

doi:10.1016/0045-7930(85)90027-1

Cited by 32 publications

(15 citation statements)

References 2 publications

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“…where £(ω, ψ) is the discrete operator of spatial derivatives including nonlinear convective terms and linear diffusive terms, and ψ is obtained from the Poisson equation. A fourth-order Runge-Kutta scheme can be written in the following form [49] ω (1) = ω n + ∆t 2 £(ω n , ψ n )…”

Section: Time Integration Algorithmsmentioning

confidence: 99%

High-order methods for decaying two-dimensional homogeneous isotropic turbulence

San

Staples

2012

Computers & Fluids

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Numerical schemes used for the integration of complex flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, the performance of various high-order accurate numerical schemes is investigated for direct numerical simulations (DNS) of homogeneous isotropic two-dimensional decaying turbulent flows. The numerical accuracy of compact difference, explicit central difference, Arakawa, and dispersion-relation-preserving schemes are analyzed and compared with the Fourier-Galerkin pseudospectral scheme. In addition, several explicit Runge-Kutta schemes for time integration are investigated. We demonstrate that the centered schemes suffer from spurious Nyquist signals that are generated almost instantaneously and propagate into much of the field when the numerical resolution is insufficient. We further show that the order of the scheme becomes increasingly important for increasing cell Reynolds number. Surprisingly, the sixth-order schemes are found to be in perfect agreement with the pseudospectral method. Considerable reduction in computational time compared to the pseudospectral method is also reported in favor of the finite difference schemes. Among the fourth-order schemes, the compact scheme provides better accuracy than the others for fully resolved computations. The fourth-order Arakawa scheme provides more accurate results for under-resolved computations, however, due to its conservation properties. Our results show that, contrary to conventional wisdom, difference methods demonstrate superior performance in terms of accuracy and efficiency for fully resolved DNS computations of the complex flows considered here. For under-resolved simulations, however, the choice of difference method should be made with care.

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Section: Time Integration Algorithmsmentioning

confidence: 99%

High-order methods for decaying two-dimensional homogeneous isotropic turbulence

San

Staples

2012

Computers & Fluids

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“…This bound was refined by Pike and Roe [13] to give ßt < m -1. Runge-Kutta methods with an odd number of stages, that attain this bound, are also secondorder accurate.…”

Section: Order Conditions and Butcher Seriesmentioning

confidence: 99%

Two-step Runge-Kutta methods and hyperbolic partial differential equations

Renaut¹

1990

Math. Comp.

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Abstract. The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80% may be achieved by using two-step Runge-Kutta methods instead of the classical onestep methods. These two-step Runge-Kutta methods were first introduced by Byrne and Lambert in 1966. They are designed to have the same number of function evaluations as the equivalent one-step schemes, and thus they are potentially more efficient. By solving a nonlinear programming problem, which is specified by stability requirements, optimal two-step schemes are designed. The optimization technique is applicable for stability regions of any shape. INTRODUCTIONIn this paper we consider a class of pseudo-Runge-Kutta methods for the solution of an ordinary differential system of equations (0.1) y' = f(y) which arises from the semidiscretization of a hyperbolic partial differential equation. In 1982, Jameson [7] initiated interest in the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. He applied the van der Houwen [4] optimal schemes in codes for the solution of the Euler equations by central differences. These schemes are optimal because they have regions of stability enclosing a maximal interval on the imaginary axis, as is required when central differences are used for the semidiscretization. Here we demonstrate that greater efficiency is achieved by using two-rather than onestep Runge-Kutta formulae. These Runge-Kutta methods were first considered by Byrne and Lambert [1] in 1966.We define an explicit two-step w-stage Runge-Kutta method as m (0.2) yn+x = (1 -ß)yn + ßyn_x + h ][>,/(>V.) + V0Í)) » i=i

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“…It can be shown (see also Pike and Roe, 1985;Ganzha and Vorozhtsov, 1993) that the left-hand side of the von Neumann stability condition (30) remains the same for RungeKutta-type schemes with a different number m of intermediate stages (m ≥ 1). Only the value of the Courant number C on the right-hand side of (30) is different for different Runge-Kutta schemes in the absence of added artificial dissipators; for example, C = 4 for the optimal five-stage Runge-Kutta scheme.…”

Section: The Three-stage Jameson Schemementioning

confidence: 96%

“…It was stressed by Pike and Roe (1985) that the Jameson's schemes belong to those rare schemes, for which it is possible to obtain the necessary von Neumann stability condition in closed form. This is extremely important for the purpose of the validation of new symbolic-numerical methods for stability analysis of numerical methods.…”

Section: Numerical Solution Of Aerodynamical Problemsmentioning

confidence: 99%

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Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

Ganzha

Vorozhtsov

1999

Journal of Symbolic Computation

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The paper deals with problems arising in the application of the computer algebra systems for the symbolic-numeric stability analysis of difference schemes and schemes of the finite-volume method approximating the two-dimensional Euler equations for compressible fluid flows on curvilinear spatial grids. We carry out a detailed comparison of the REDUCE 3.6 and Mathematica (Versions 2.2 and 3.0) from the point of view of their applicability to the solution of the above problems. We draw a conclusion that a preference should be given for Mathematica from the viewpoint of the execution of symbolic-numeric computations. We also describe in detail our new symbolic-numeric algorithm for stability investigation, which was implemented with the aid of Mathematica. The proposed method enables us to reduce the needed computer storage at the symbolic stages by a factor of about 20 as compared with the previous algorithms. A feature of the numerical stages is the use of the arithmetic of rational numbers, which enables us to avoid the accumulation of the roundoff errors. We present the examples of the application of the proposed symbolic-numeric method for stability analysis of very complex schemes of the finite-volume method on curvilinear grids, which are widely used in computational fluid dynamics.

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Accelerated convergence of Jameson's finite-volume Euler scheme using van der Houwen integrators

Cited by 32 publications

References 2 publications

High-order methods for decaying two-dimensional homogeneous isotropic turbulence

High-order methods for decaying two-dimensional homogeneous isotropic turbulence

Two-step Runge-Kutta methods and hyperbolic partial differential equations

Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

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