Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids

Park, Jin Seok; Yoon, Sung-Hwan; Kim, Chongam

doi:10.1016/j.jcp.2009.10.011

Cited by 151 publications

(96 citation statements)

References 45 publications

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“…At the same time, it is also observed that well-controlled vertex value at interpolation/limiting stage makes it possible to produce monotonic distribution of cell-averaged values. Extensive numerical experiments [20][21][22] strongly support that full realization of Eq. (16) is very effective to preserve accurate monotone profiles.…”

Section: A Mlp-u Slope Limitermentioning

confidence: 64%

“…S Tj is called the MLP stencil. 21,22 It is noted that the MLP condition can be applied to any type of mesh since it does not assume particular mesh connectivity. At the same time, it is also observed that well-controlled vertex value at interpolation/limiting stage makes it possible to produce monotonic distribution of cell-averaged values.…”

Section: A Mlp-u Slope Limitermentioning

confidence: 99%

“…21,22 For higher-order polynomial approximations greater than P1 approximation, however, additional condition is necessary because the P1 -based MLP condition may omit some troubled cells that violate the maximum principle.…”

Section: B Augmented Mlp Conditionmentioning

confidence: 99%

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Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Park

Chang

Kim

2014

52nd Aerospace Sciences Meeting

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Higher-order multi-dimensional limiting Process (MLP) [J. S. Park and C. Kim, Higherorder Multi-dimensional Limiting Strategy for Discontinuous Galerkin Methods in Compressible Inviscid and Viscous Flows, Comp. & Fluids, In press] is improved and applied to correction procedure via reconstruction (CPR) on unstructured grids. MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn reconstruction. The resulting algorithms, called the hierarchical MLP, facilitate the accurate capturing of detailed flow structures in both continuous and discontinuous regions. This algorithm has been developed in the modal DG framework, but it also can be formulated into a nodal framework, most notably the CPR framework. Troubled-cells are detected by applying the MLP concept, and the final accuracy is determined by the projection procedure and MLP limiting step. Through extensive numerical analyses and computations, it is demonstrated that the proposed limiting approach yields the desired accuracy and outstanding performances in resolving compressible inviscid and viscous flow features.

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Section: A Mlp-u Slope Limitermentioning

confidence: 64%

Section: A Mlp-u Slope Limitermentioning

confidence: 99%

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Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Park

Chang

Kim

2014

52nd Aerospace Sciences Meeting

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“…For turbulence prediction the RANS approach is implemented along with appropriate statistical two-equation models, namely the k − ε [30], the k − ω [31] and the SST (Shear Stress Transport) [32] models. An upwind method is implemented for the computation of the inviscid fluxes, applying the Roe's [33] or the HLLC (Harten-Lax-van Leer-Contact) [34] approximate Riemann solvers, coupled with a higher-order accurate spatial scheme, based on the MUSCL (Monotone Upwind Scheme for Conservation Laws) approach along with appropriate slope limiters, namely the Van Albada-Van-Leer [35], the Min-mod [36], the Barth-Jespersen [37] and the MLP-Venkatakrishnan (Multi-dimensional Limiting Process-Venkatakrishnan) [38,39] limiter. For the computation of the viscous fluxes the required velocity and temperature gradients are evaluated with an element-based (edge-dual volume) or a nodal-averaging method [28], either selected by the user.…”

Section: The Galatea Codementioning

confidence: 99%

Numerical Analysis of Rarefied Gas Flows Using the Academic CFD Code Galatea

Klothakis¹,

Lygidakis²,

Nikolos³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. During the last decades considerable efforts have been exerted for the development of micro air vehicles as well as microelectromechanical systems in general, for a wide range of applications. However, such systems involve microscale rarefied gas flows, which appear to be significantly different comparing to flows at macroscale and continuum regime; it is this the reason the Navier-Stokes equations fail to simulate such phenomena without further modification. To this end, the enhancement of the in-house academic Computational Fluid Dynamics solver Galatea to encounter such simulations is reported in this study. In case of rarefied gas flows and particularly for fluids in slip flow regime (Knudsen number greater than 0.01) the no-slip condition on solid wall surfaces is no longer valid; hence, velocity slip conditions as well as temperature jump ones have to be included instead. Furthermore, to increase accuracy at the same region the second-order accurate spatial slip model of Beskok and Karniadakis has been incorporated, which avoids the numerical difficulties, entailed by the evaluation of the second derivative of slip velocity when complex geometries along with unstructured hybrid grids are encountered. Due to oscillations that might appear, especially during the initial steps of the iterative procedure, a normalization scheme is additionally employed, to allow for the gradual increase of the corresponding slip/jump values. Galatea has been validated against a benchmark test case concerning rarefied laminar flow (inside the slip flow regime) over a wing with a NACA0012 airfoil in different angles of attack. The obtained results were compared with those of a reference solver, and with those obtained with the paralleld open-source kernel SPARTA, based on the Direct Simulation Monte-Carlo method. According to this last approach, the flow domain is divided into a finite number of computational cells, while the required sample macroscopic flow properties are retrieved assuming intermolecular collisions of the simulated particles inside such cells. An excellent agreement was achieved between the results obtained by Galatea and SPARTA as well.

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“…For example, Batten et al [6] construct several trial gradients, limit each using a scalar limiter, and then choose the one with the largest norm. Park et al [28] extended the multi-dimensional limiting process (MLP) introduced in [24,33] from structured to unstructured meshes, and use an enlarged stencil in the limiting process.…”

mentioning

confidence: 99%

Two-Dimensional Slope Limiters for Finite Volume Schemes on Non-Coordinate-Aligned Meshes

May¹,

Berger²

2013

SIAM J. Sci. Comput.

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Abstract. In this paper we develop a new limiter for linear reconstruction on non-coordinatealigned meshes in two space dimensions, with focus on Cartesian embedded boundary grids. Our limiter is inherently two-dimensional and linearity preserving. It separately limits the x and y components of the gradient, as opposed to a scalar limiter which limits all components simultaneously with one scalar. The limiter is based on solving a tiny linear program (LP) on each cell, using a very efficient version of the simplex method. A variety of computational results on triangular and embedded boundary meshes are presented. They demonstrate that the LP limiter successfully removes oscillations and significantly increases solution accuracy compared to a scalar limiter.

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Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids

Cited by 151 publications

References 45 publications

Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Numerical Analysis of Rarefied Gas Flows Using the Academic CFD Code Galatea

Two-Dimensional Slope Limiters for Finite Volume Schemes on Non-Coordinate-Aligned Meshes

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