 # 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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Numerical Analysis Physical Sciences > Modeling and Simulation Physical Sciences > Physics and Astronomy (miscellaneous) Physical Sciences > General Physics and Astronomy Physical Sciences > Computer Science Applications Physical Sciences > Computational Mathematics Physical Sciences > Applied Mathematics English 5 February 2015 03 Feb 2016 10:32 30 Jul 2020 21:14 Elsevier 0021-9991 Green https://doi.org/10.1016/j.jcp.2014.11.031 : FunderFP7: Grant ID226316: Project TitleADDECCO - Adaptive Schemes for Deterministic and Stochastic Flow Problems: FunderFP7: Grant ID265780: Project TitleIDIHOM - Industrialisation of High-Order Methods â�� A Top-Down Approach

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