Efficient Relaxation Methods for High-Order Discretization of Steady Problems

May, Georg; Jameson, Antony

doi:10.1142/9789814313193_0013

Cited by 7 publications

(8 citation statements)

References 46 publications

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“…In order to speed up the convergence, one can use other relaxation methods, which are developed specifically for integrating steady-state equations ( e.g. May and Jameson 2011). They often involve a relaxation variable which is called ‘pseudo time’.…”

Section: Introductionmentioning

confidence: 99%

Stationary relativistic jets

2015

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In this paper we describe a simple numerical approach which allows to study the structure of steady-state axisymmetric relativistic jets using one-dimensional time-dependent simulations. It is based on the fact that for narrow jets with v z ≈ c the steady-state equations of relativistic magnetohydrodynamics can be accurately approximated by the one-dimensional time-dependent equations after the substitution z = ct. Since only the time-dependent codes are now publicly available this is a valuable and efficient alternative to the development of a high-specialised code for the time-independent equations. The approach is also much cheaper and more robust compared to the relaxation method. We tested this technique against numerical and analytical solutions found in literature as well as solutions we obtained using the relaxation method and found it sufficiently accurate. In the process, we discovered the reason for the failure of the self-similar analytical model of the jet reconfinement in relatively flat atmospheres and elucidated the nature of radial oscillations of steady-state jets.

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Section: Introductionmentioning

confidence: 99%

Stationary relativistic jets

2015

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“…Various approaches are used in practice, including NewtonKrylov methods such as GMRES, block Jacobi methods, line Jacobi methods, Gauss-Seidel (GS) methods, symmetric GS (SGS) methods, and lower-upper SGS (LU-SGS) methods (developed by Jameson and Yoon [68] and Yoon and Jameson [145]). For full details of all aforementioned approaches the reader is referred to the review of Wang [139], and the book chapter of May and Jameson [90]. However, we will note two important points here.…”

Section: Implicit Methodsmentioning

confidence: 99%

“…This topic is discussed in some detail within the general review article of Wang [139]. More recently, reviews specifically on this topic by Iacono and May [60] and Iacono, May and Wang [61] have been presented, along with a book chapter by May and Jameson [90]. Based on this literature, it is apparent that two major issues are inhibiting the efficient temporal integration of unstructured high-order spatial discretizations.…”

Section: Time Integration 331 Overviewmentioning

confidence: 99%

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Vincent

Jameson

2011

Math. Model. Nat. Phenom.

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Abstract. Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of high-order methods amongst a non-specialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat towards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

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“…As has been realized by many researchers [1,2], one big challenge in the development of high-order method is to design a robust, scalable and efficient algorithm for solving the nonlinear system which governs complex fluid flow, e.g., high Reynolds number turbulent flow over complex geometry. For steady flow simulation, mature convergence acceleration techniques [3,4,5,6,7] have been established for low-order finite volume methods.…”

Section: Introductionmentioning

confidence: 99%

“…For steady flow simulation, mature convergence acceleration techniques [3,4,5,6,7] have been established for low-order finite volume methods. However, these techniques cannot be directly carried over to highorder schemes [2]. Much effort [8,9,10,11,12,13,14] has been spent on developing robust and efficient solution algorithm for high-order schemes.…”

Section: Introductionmentioning

confidence: 99%

Homotopy Continuation for Correction Procedure via Reconstruction - Discontinuous Galerkin (CPR-DG) Methods

Wang

2015

53rd AIAA Aerospace Sciences Meeting

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Homotopy continuation for correction procedure via reconstruction -discontinuous Galerkin (CPR-DG) methods is developed for solving steady state hyperbolic conservation laws. The efficacy of homotopy continuation has been demonstrated using both 1D and 2D steady flow problems. According to the test results, it is found that homotopy continuation can work robustly on flow simulations of shock capturing for a wide range of grid sizes and polynomial orders without parameter tuning.

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Efficient Relaxation Methods for High-Order Discretization of Steady Problems

Cited by 7 publications

References 46 publications

Stationary relativistic jets

Stationary relativistic jets

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Homotopy Continuation for Correction Procedure via Reconstruction - Discontinuous Galerkin (CPR-DG) Methods

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