Higher Order Cell Centered Finite Volume Schemes for Unstructured Cartesian Grids

Dement, David C.; Ruffin, Stephen M.

doi:10.2514/6.2018-1305

Cited by 9 publications

(17 citation statements)

References 1 publication

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“…However, many current schemes in the flux-balance form are based on the MUSCL approach, where the numerical flux is evaluated with high-order reconstructed solutions. As shown in the previous articles 2, 3 and also pointed out in References 14,15, such schemes can be second-order at best for nonlinear equations although still bring improvements to complex flow simulations as demonstrated in References 5,16,17. In this article, we will focus on schemes that can be genuinely high-order on regular grids, which are strongly desired for scale-resolving turbulent-flow simulations requiring highly refined grids (where high-order schemes are more efficient than second-order schemes). As we discussed in the previous article 3 and will also discuss later, genuine high-order accuracy requires flux reconstruction since a scheme must be finite-difference.…”

Section: Introductionmentioning

confidence: 80%

“…Reference 35 mentions the possibility of achieving high-order accuracy via reconstruction of the function whose cell average is the point-valued flux, but it is in fact true only for uniform grids as we will discuss later in Section 3.6. Some existing unstructured-grid methods based on high-order solution reconstruction schemes 5,16,17 are described as discretizations of Equation (14). This is confusing because they present high-order accurate results, which contradicts the statement above.…”

Section: Point-valued Differential Form With An Approximate Flux Integralmentioning

confidence: 99%

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Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

Nishikawa

2021

Numerical Methods in Fluids

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In this article, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended -scheme combined with economical flux reconstruction formulas, achieving up to fifth-order accuracy (sixth-order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.

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Section: Introductionmentioning

confidence: 80%

Section: Point-valued Differential Form With An Approximate Flux Integralmentioning

confidence: 99%

Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

Nishikawa

2021

Numerical Methods in Fluids

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“…[36,37]. Although there are high-order accuracy verification studies seen in the literature [11,32,38], these high-order results are due to unexpected linearization of nonlinear equations by a particular class of exact solutions (i.e., a function with a small perturbation). Later, we will show an example of this phenomenon.…”

Section: Finite-volume With Point-valued Solution (Quick): Third-order Accurate Withmentioning

confidence: 99%

“…Therefore, all the existing U-MUSCL schemes [4,11,32,38] are second-order accurate at best for nonlinear equations such as the Euler equations even on a regular grid. Nevertheless, these schemes have been demonstrated to provide improved resolution (see also Ref.…”

Section: Remarksmentioning

confidence: 99%

Resolving Confusion Over Third-Order Accuracy of U-MUSCL

Padway

Nishikawa

2021

AIAA Scitech 2021 Forum

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In this paper, we discuss the U-MUSCL reconstruction scheme -an unstructured-grid extension of Van Leer's κ-scheme -proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999]. This technique has been widely used in practical unstructured-grid fluid-dynamics solvers but with confusions: e.g., third-order accuracy with κ = 1/2 or κ = 1/3. This paper clarifies some of these confusions: e.g., the U-MUSCL scheme can be third-order accurate in the point-valued solution with κ = 1/3 on regular grids for linear equations in all dimensions, it can be third-order accurate with κ = 1/2 as the QUICK scheme in one dimension. It is shown that the U-MUSCL scheme cannot be third-order accurate for nonlinear equations, except a very special case of κ = 1/2 on regular simplex-element grids, but it can be an accurate low-dissipation second-order scheme. It is also shown that U-MUSCL extrapolates a quadratic function exactly with κ = 1/2 on arbitrary grids provided the gradient is computed by a quadratic least-squares method. Two techniques are discussed, which transform the U-MUSCL scheme into being genuinely third-order accurate on a regular grid: an efficient flux-reconstruction method and a special source term quadrature formula for κ = 1/2.

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“…There is a class of numerical schemes that are high-order accurate for linear equations but second-order accurate for nonlinear equations (e.g., those in Ref. [10,11,12,13,14,15]), which we may call linearly high-order schemes. As pointed out in Refs.…”

Section: Introductionmentioning

confidence: 99%

Implicit gradients based conservative numerical scheme for compressible flows

Chamarthi¹,

Hoffmann²,

Nishikawa³

et al. 2021

Preprint

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This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cellcentered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new approach has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method's capability using the implicit gradient method for compressible flows.

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Higher Order Cell Centered Finite Volume Schemes for Unstructured Cartesian Grids

Cited by 9 publications

References 1 publication

Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

Resolving Confusion Over Third-Order Accuracy of U-MUSCL

Implicit gradients based conservative numerical scheme for compressible flows

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