High order residual distribution scheme for Navier-Stokes equations

Abgrall, Rémi; Santis, Dante De

doi:10.2514/6.2011-3231

Cited by 10 publications

(8 citation statements)

References 31 publications

(33 reference statements)

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“…We do not consider high-order (curved) elements, e.g., curved P 2 elements, because the third-order scheme considered here relies on a special property of the edgebased scheme that it achieves third-order accuracy on linear triangles, which would probably be lost on curved elements. Compared with other high-order methods, which fail to achieve the design order of accuracy without curved elements such as the discontinuous Galerkin method [27,28] and the continuous finite-element-type methods [29][30][31], the thirdorder scheme considered here does not require boundary conditions imposed over the boundary edge (they are imposed at nodes), nor extra degrees of freedom placed between two boundary nodes, which should be located at the physical boundary and thus makes a curved boundary edge inevitable on a curved boundary. We point out that similar third-order schemes exist that preserve third-order accuracy without curved elements [32,33].…”

Section: Potential Flow Over Curved Boundarymentioning

confidence: 98%

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

Nishikawa

2015

Journal of Computational Physics

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Section: Potential Flow Over Curved Boundarymentioning

confidence: 98%

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

Nishikawa

2015

Journal of Computational Physics

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“…The total residual is first computed without considering the boundary contributions, then a correction residual is added to correctly take into account the boundary conditions. For a node i belonging to the boundary, the residual associated to the boundary conditions can be written as [8] …”

Section: 4mentioning

confidence: 99%

High-Order Preserving Residual Distribution Schemes for Advection-Diffusion Scalar Problems on Arbitrary Grids

Abgrall¹,

Santis²,

Ricchiuto³

2014

SIAM J. Sci. Comput.

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This paper deals with the construction of a class of high-order accurate residual distribution schemes for advection-diffusion problems using conformal meshes. The problems considered range from pure diffusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is reconstructed at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and nonlinear schemes are constructed and their accuracy is tested with the discretization of advectiondiffusion and anisotropic diffusion problems. Abstract. This paper deals with the construction of a class of high order accurate Residual Distribution schemes for advection-diffusion problems using conformal meshes. The problems considered range from pure difusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradients of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and non-linear schemes are constructed and their accuracy is tested with the discretization of advection-diffusion and anisotropic diffusion problems.1. Introduction. In the last years different high order schemes have been developed to obtain an higher order (more than two) discretization of the Navier-Stokes equations. One of the most attractive scheme seems to be the discontinuous Galerkin (DG) scheme [12]. Residual Distribution (RD) schemes [23, 1, 3] represent a very interesting alternative to DG schemes. While computationally compact and probably more flexible, DG schemes suffer from the serious drawback of a very fast growth of the number of degrees of freedom (DOF) with the cell polynomial degree. In RD schemes the formulation remains local, as in DG, but the number of DOFs growths less quickly because the solution is assumed to be continuous. Another difference between RD and DG schemes is that, to date at least, the non oscillatory properties of the RD scheme in the case of discontinuous solutions are probably better understood that for their DG counterpart.RD schemes have been developed mainly for advection problems due to possibility to construct multidimensional upwind schemes which guarantees a small disc...

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“…We describe this in the steady 2D case, the extension to the unsteady case or the 1D case is quite straightforward. In our simulation, we have chosen to use the scheme developed in related works, but this is not essential and what matters is that the stencil of the method is relatively compact because we are integrating a PDE of the form…”

Section: Computational Costmentioning

confidence: 99%

“…Representation of the set  i . The vertex M i is indicated with a • and the elements of  i by • related works,[52][53][54] but this is not essential and what matters is that the stencil of the method is relatively compact because we are integrating a PDE of the form(12) …”

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confidence: 99%

Model reduction using L¹‐norm minimization as an application to nonlinear hyperbolic problems

Abgrall

Crisovan

2018

Numerical Methods in Fluids

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Summary We are interested in the model reduction techniques for hyperbolic problems, particularly in fluids. This paper, which is a continuation of an earlier paper of Abgrall et al, proposes a dictionary approach coupled with an L1 minimization approach. We develop the method and analyze it in simplified 1‐dimensional cases. We show in this case that error bounds with the full model can be obtained provided that a suitable minimization approach is chosen. The capability of the algorithm is then shown on nonlinear scalar problems, 1‐dimensional unsteady fluid problems, and 2‐dimensional steady compressible problems. A short discussion on the cost of the method is also included in this paper.

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High order residual distribution scheme for Navier-Stokes equations

Cited by 10 publications

References 31 publications

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

High-Order Preserving Residual Distribution Schemes for Advection-Diffusion Scalar Problems on Arbitrary Grids

Model reduction using L¹‐norm minimization as an application to nonlinear hyperbolic problems

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High order residual distribution scheme for Navier-Stokes equations

Cited by 10 publications

References 31 publications

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

High-Order Preserving Residual Distribution Schemes for Advection-Diffusion Scalar Problems on Arbitrary Grids

Model reduction using L1‐norm minimization as an application to nonlinear hyperbolic problems

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Model reduction using L¹‐norm minimization as an application to nonlinear hyperbolic problems