Header

UZH-Logo

Maintenance Infos

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


Abgrall, Rémi; de Santis, Dante (2015). Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier–Stokes equations. Journal of Computational Physics, 283:329-359.

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 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 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.

Statistics

Citations

7 citations in Web of Science®
11 citations in Scopus®
Google Scholar™

Altmetrics

Downloads

14 downloads since deposited on 03 Feb 2016
9 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:5 February 2015
Deposited On:03 Feb 2016 10:32
Last Modified:08 Dec 2017 21:05
Publisher:Elsevier
ISSN:0021-9991
Publisher DOI:https://doi.org/10.1016/j.jcp.2014.11.031

Download

Download PDF  'Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier–Stokes equations'.
Preview
Content: Accepted Version
Language: English
Filetype: PDF
Size: 6MB
View at publisher