A central Rankine–Hugoniot solver for hyperbolic conservation laws

Jaisankar, S.; Rao, S. V. Raghurama

doi:10.1016/j.jcp.2008.10.002

Cited by 32 publications

(41 citation statements)

References 37 publications

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“…Now, if we compare with more recent finite-volume methods, HLL-2D gives numerical results that are equivalent with those of [10] or [11], for example, with the same grid resolution for [10] and a grid resolution slightly lower for [11].…”

Section: Examplementioning

confidence: 92%

“…n ext ) dl = j j ( F(U ). n ext ) dl (10) In its high accuracy version, the line integral from Equation (10) is discretized using a two-point Gaussian integration formula. This formula is exact for a polynomial of degree up to 3.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…G. CAPDEVILLE distribution approach was exclusively designed for marching to a steady solution. Only recently, there have been some attempts in formulating time accurate extensions [7] or extending residual distribution schemes to quadrilateral grids [8].On the other side, the finite-volume approach is more flexible; see for example the recent publications [9][10][11] or the proceedings [12] for more references on this issue.First, the classical 'method of lines' makes it possible a separated optimization of the temporal and spatial discretizations. Second, there exist many possibilities to introduce some multidimensional information into the Riemann solvers used in conventional upwind finite-volume schemes: multi-directional methods or minimum-strength wave models constitute possible choices [5].…”

mentioning

confidence: 99%

“…On the other side, the finite-volume approach is more flexible; see for example the recent publications [9][10][11] or the proceedings [12] for more references on this issue.…”

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confidence: 99%

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A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

Capdeville

2010

Numerical Methods in Fluids

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SUMMARYThis article presents a numerical model that enables to solve on unstructured triangular meshes and with a high order of accuracy, Riemann problems that appear when solving hyperbolic systems.For this purpose, we use a MUSCL-like procedure in a 'cell-vertex' finite-volume framework.In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the barycenter of a triangular cell, from the surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations.In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we use and adapt the monotonicity-preserving limiter, initially devised by Barth (AIAA Paper 90-0013, 1990).Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model.

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Section: Examplementioning

confidence: 92%

Section: Spatial Discretizationmentioning

confidence: 99%

mentioning

confidence: 99%

“…On the other side, the finite-volume approach is more flexible; see for example the recent publications [9][10][11] or the proceedings [12] for more references on this issue.…”

mentioning

confidence: 99%

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A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

Capdeville

2010

Numerical Methods in Fluids

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“…In a ducted flow, the effect of pressure and velocity are contrary to each other, hence pressure rise at any point in the flow is accompanied by a corresponding decrease in velocity. At very high pressure and density prevailing in the duct, the expanding shock wave may cause localized subsonic pockets of flow to occur near the point of shock reflection and hence the appearance of Mach Figure 2 Abrupt Pressure Rise across Shock Figure 3 Viscid and Inviscid Discontinuities stem [7,8]. Numerically, this abrupt pressure rise at reflection point O in Figure 2 can be controlled by stencil averaging [9,10] of density, ρ or Pressure,P and by ensuring local flow velocity, u in the streamwise direction to be fully supersonic so that only regular shock reflection occurs at the wall.…”

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confidence: 99%

Development of meshless method boundary conditions for high speed flows

2017

IJLTET

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SHOCK STRUCTURE MODELDuring numerical flow simulation, diffusion can be induced through artificial viscosity term to control development of flow. Early researchers included the dissipation term in the formulation of finite volume Euler equations to suppress odd-even point oscillations arising from pressure-velocity coupling. It was found that second difference diffusion terms with second and fourth difference pressure coefficients rendered wiggle-free computation. The highorder central differencing schemes produce accurate results in smooth flow regions but give rise to oscillations at shocks or discontinuities. This problem has been resolved through special upwinding methods and by Roe linearization technique [1], with or without characteristic decomposition. The development of Euler codes has paralleled the development of positivity or monotonicity preserving schemes that satisfy Jameson's local extremum diminishing principle (LED criterion) [2] in order that physically meaningful values of pressure and density could be computed. Customarily, the residual is evaluated as convective and diffusive fluxes separately in order to improve the accuracy of computation of flow properties. If meshless technique is employed then the diffusive flux can be obtained as the product of point fluxes and least square (LSQ) coefficients along coordinate directions prior to time integration to steady state solution. Irregular boundaries and fine broken cells at the wall, which were constraints to computational accuracy of Cartesian mesh solvers, are now implemented efficiently by meshless wall boundary techniques [3]. The accuracy with which shock waves reflecting at the wall boundaries can be captured depends both on the computational schemes used and the shock structure model adopted. Even after a century old research, a concrete shock wave development model does not exist. Among various models suggested by researchers, the three-shock theory [4, 5] (3ST) putforth by von Neumann stands out as the most realistic one to describe the commonly encountered shock wave phenomena. In situations where only a regular shock reflection is expected as in ducted flows, a Mach wave reflection too may occur. A consequence of this effect is the appearance of Mach stem at the wall accompanied by a regular shock wave. Now, according to 3ST, non-regular shock reflection called Mach reflection (MR) is produced by three intersecting shock waves: incident wave, reflected wave and Mach stem. Their point of intersection called triple point (T) may move toward, away or parallel to the reflecting surface, giving rise to direct, inverse or stationary MR as shown in the Figures 1. These occurrences are termed as Mach stem and it forms the subject matter of this paper.

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Local maximum principle satisfying high‐order non‐oscillatory schemes

Dubey

Biswas

Gupta

2015

Numerical Methods in Fluids

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Summary The main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.

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A central Rankine–Hugoniot solver for hyperbolic conservation laws

Cited by 32 publications

References 37 publications

A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

Development of meshless method boundary conditions for high speed flows

Local maximum principle satisfying high‐order non‐oscillatory schemes

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