Numerical approximation of parabolic problems by residual distribution schemes

Abgrall, Rémi; Baurin, Guillaume; Krust, Arnaud; Santis, Dante De; Ricchiuto, Mario

doi:10.1002/fld.3710

Cited by 9 publications

(11 citation statements)

References 26 publications

(65 reference statements)

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“…The objective is to show that (i) the high order RD schemes previously proposed can be successfully used in the discretization of the advection-diffusion equation, (ii) the high order accuracy is preserved in all the range of the Peclet number. This is contrast with the method proposed in [5] where the region P e ≈ 1 was problematic.…”

Section: Strategy (A)contrasting

confidence: 45%

“…A more general scheme consists in using distribution coefficients which are function of the local Peclet number in order to recover an isotropic scheme in the diffusion limit and an upwind scheme in the advection limit [19,11]. Another attempt in that direction is given by [5], the scheme give satisfactory results except in the region P e ≈ 1, which is typical of a boundary layer. Hence the present contribution can be viewed as an improvement over the previous references.…”

Section: Consistency and Accuracymentioning

confidence: 99%

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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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Section: Strategy (A)contrasting

confidence: 45%

Section: Consistency and Accuracymentioning

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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“…On the other hand, this means that better solutions to extend RD to viscous flows have to be sought. This justifies the recent investment in genuinely residual based approximations of the viscous terms by H. Nishikawa [21,22], as well as by some of the authors of the present work [3,4,5]. While this new approach might allow a genuine improvement over the hybrid RD-Galerkin formulation.…”

Section: Discussionsupporting

confidence: 63%

“…Our results show that, while in the scalar case one can clearly show that this approach allows to recover a uniform second order convergence rate, for moderate/high Reynolds laminar flows its beneficial effects are much less pronounced even when looking at friction coefficients. This justifies the quest for new consistent approaches allowing to further reduce the discretization error in viscous flows [21,22,3,4,5].…”

mentioning

confidence: 99%

On hybrid residual distribution–Galerkin discretizations for steady and time dependent viscous laminar flows

Dobeš

Ricchiuto

Abgrall

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in Ricchiuto (2008) and Villedieu etal. (2011), we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in Ricchiuto etal. (2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar Navier-Stokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show that in practice, the use of Peclet number based corrections does not change dramatically the quality of the solutions obtained. This justifies the quest for new formulations. This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in (Ricchiuto et al. J.Comp.Appl.Math. 215, 2008, Villedieu et al. J.Comput.Phys, 230, 2011, we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in (Ricchiuto et al. J.Comp.Appl.Math. 215, 2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar NavierStokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show t...

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“…It has been shown [3] that a scheme resulting from the coupling of a RD scheme with a Galerkin discretization of the viscous term is still a residual method, but this is true only for P1 elements. Furthermore it is well know [26] that the scheme obtained as the sum of the RD scheme and the Galerkin scheme is second order accurate in the diffusion and advection limits, but it is only first order accurate when advection and diffusion are equally important.…”

Section: Extension To Viscous Termsmentioning

confidence: 99%

High order residual distribution scheme for Navier-Stokes equations

Abgrall

Santis

2011

20th AIAA Computational Fluid Dynamics Conference

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In this work we describe the use of the Residual Distribution schemes applied to the discretization of conservation laws. In particular, emphasis is put on the construction of a third order accurate scheme. We first recall the properties of a Residual Distribution scheme and we show how to construct a high order scheme for advection problems. Furthermore, we show how to speed up the convergence of the implicit scheme to the steady solution by the means of the Jacobian-free technique. We then extend the scheme to the case of advection-diffusion problems. In particular, we propose a new approach in which the residuals of the advection and diffusion terms are distributed together to get high order accuracy. Due to the continuous approximation of the solution, the gradients of the variables are reconstructed at the nodes and then interpolated on the elements. The numerical scheme is used to discretize the advection-diffusion scalar problem and the compressible Navier-Stokes equations.

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Numerical approximation of parabolic problems by residual distribution schemes

Cited by 9 publications

References 26 publications

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

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

On hybrid residual distribution–Galerkin discretizations for steady and time dependent viscous laminar flows

High order residual distribution scheme for Navier-Stokes equations

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