On uniformly high-order accurate residual distribution schemes for advection–diffusion

2012

Commun. comput. phys.

Abstract. We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives.

Section: A Simple Formulationmentioning

confidence: 99%

A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art

2012

Commun. comput. phys.

“…To address this problem mixed schemes have been developed, in which RD methods for the advection terms are combined with the Galerkin discretization of the diffusion terms. For such type of schemes, a proper blending between the RD and the Galerkin schemes must be constructed otherwise the accuracy of the resulting schemes is spoiled when advection and diffusion are equally important [25,28].…”

Section: Introductionmentioning

confidence: 99%

Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier–Stokes equations

Journal of Computational Physics

Santis

2015

Self Cite

A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier-Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non-linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of 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, preserving the optimal accuracy of the method. Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions. Abstract A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier-Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each...

“…As remarked in [23,29], this approach is not satisfactory because its accuracy is not uniform over the whole range of values of the mesh size, viscosity, local speeds. In particular, while on relatively coarse meshes second order is indeed obtained in practice, as one consider finer meshes only first order rates are observed.…”

mentioning

confidence: 99%

“…In this method, RD is re-cast as a stabilized Galerkin discretization. By analogy with what is done in the SUPG and GLS case [19,18], in [29,37] a dependence of the stabilization term on the cell Peclet/Reynolds number is introduced so that in diffusion dominated regions the full Galerkin scheme is recovered, while the hybrid method is used in high Reynolds regions.…”

mentioning

confidence: 99%

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

Dobeš

Ricchiuto

Computer Methods in Applied Mechanics and Engineering

et al. 2015

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...