doi:10.1016/j.paerosci.2007.05.001 
Copyright © 2007 Elsevier Ltd All rights reserved.
High-order methods for the Euler and Navier–Stokes equations on unstructured grids
Z.J. Wang
, a, 
aDepartment of Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, IA 50011, USA
Available online 19 July 2007.
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Abstract
This article reviews several unstructured grid-based high-order methods for the compressible Euler and Navier–Stokes equations. We treat the spatial and temporal discretizations separately, hoping that it is easier to spot the similarities and differences of various types of methods. Our main focus is to present the basic design principles of each method, and highlight its pros and cons when appropriate. Sample computational results are shown to illustrate the capability of selected methods. These high-order methods are expected to be more efficient than low-order methods for problems requiring high accuracy, such as wave propagation problems, vortex-dominated flows including high-lift configuration, helicopter blade vortex interaction, as well as large eddy simulation and direct numerical simulation of turbulence. We conclude the paper with several current challenges in the proliferation of high-order methods in the aerospace community.
Keywords: High-order; Unstructured grids; CFD; Navier–Stokes; Euler equations
Fig. 1. Illustration of algebraic (or fixed order) convergence with h-refinement, and spectral (or exponential) convergence with p-refinement, where NDOFs is the total number of degrees of freedom, and d is the space dimension. (a) Log–log scale, (b) linear–log scale.
Fig. 2. A generic error vs. cost plot for low- and high-order methods to determine which method should be chosen for certain applications.
Fig. 3. Candidate reconstruction stencils in an ENO reconstruction.
Fig. 4. Computed density contours using WENO schemes for the double Mach reflection problem (courtesy of Hu and Shu [53]). (a) Third-order WENO (1.28 M DOFs), (b) fourth-order WENO (1.28 M DOFs).
Fig. 5. Close-up view of the density contours for the double Mach reflection problem (courtesy of Hu and Shu [53]). (a) Third-order WENO (1.28 M DOFs), (b) fourth-order WENO (1.28 M DOFs).
Fig. 6. Computed density contours using DG schemes for the double Mach reflection problem (courtesy of Cockburn and Shu [73]). (a) Second-order DG, 
(2.76 million DOFs), (b) third-order DG, 
(1.38 million DOFs), (c) third-order DG, 
(5.53 million DOFs).
Fig. 7. Close-up view of the density contours computed with DG schemes for the double Mach reflection problem (courtesy of Cockburn and Shu [73]). (a) Third-order DG, 
, (1.38 million DOFs), (b) second-order DG, 
, (2.76 million DOFs), (c) third-order DG, 
, (5.53 million DOFs).
Fig. 8. Inviscid flow over an NACA0012 airfoil (courtesy of Venkatakrishnan et al. [69]).
Fig. 9. Inviscid flow over a Joukowski airfoil (courtesy of Venkatakrishnan et al. [69]).
Fig. 10. High Reynolds number suction boundary layer, Re=106 (courtesy of Venkatakrishnan et al. [69]).
Fig. 11. Global view of the grid around turbine blades (courtesy of Bassi et al. [93]).
Fig. 12. Snapshot of the turbulent intensity using third-order DG (courtesy of Bassi et al. [93]).
Fig. 13. Turbine blade density fields during a vortex shedding cycle, third-order DG (courtesy of Bassi et al. [93]).
Fig. 14. One target and two target cases for the N scheme.
Fig. 15. A quadratic element in the RD method showing six DOFs.
Fig. 16. Computational mesh for the Burger's equation (courtesy of Abgrall and Roe [103]).
Fig. 17. Isolines of the exact and computed solutions for the Burger's equations using high-order RD schemes (courtesy of Abgrall and Roe [103]).
Fig. 18. Partitions of various orders in a triangular spectral volume, the third- and fourth-order partitions are found by Van den Abeele and Lacor [116]. (a) Second-order, (b) third-order, (c) fourth-order.
Fig. 19. Nodal sets in a triangular SV supporting quadratic, cubic and quartic data reconstructions for the flux vector, shown in (a), (b) and (c), respectively.
Fig. 20. General third- and fourth-order SV partitions. (a) Third-order, (b) fourth-order.
Fig. 21. Density contours computed using SV schemes with the Rusanov flux and TVD limiter, 30 even contours between 1.25 and 21.5 [107]. (a) Second-order SV, 
(392,384 DOFs), (b) third-order SV, 
(197,616 DOFs), (c) third-order SV, 
(784,968 DOFs).
Fig. 22. Close-up view of the density contours near the double Mach stem [107]. (a) Second-order SV, 
(392,384 DOFs), (b) third-order SV, 
(197,616 DOFs), (c) third-order SV, 
(784,968 DOFs).
Fig. 23. Solution (solid circles) and flux points (solid squares) for first, second and third-order SD schemes. (a) First order, (b) second order, (c) third-order.
Fig. 24. Illustration of flux computation for face and corner points.
Fig. 25. Distribution of solution points (circles) and flux points (squares) in standard element for a third-order SD scheme.
Fig. 26. Computational grids for subsonic flow around a NACA0012 airfoil [120]. (a) Second-order FV grid, (b) third-order SD grid.
Fig. 27. Comparison of entropy error along the airfoil surface [120].
Fig. 28. Computed pressure (on the sphere) and Mach number (on z=0 plane) distributions using the fourth- and sixth-order SD schemes. Mach contours start at 
with a 
interval [128].
Fig. 29. Comparison of streamlines of the flow field between computation and experiment [128]. (a) Computation, (b) experiment.
Fig. 30. Predicted skin friction coefficient profiles with the fourth- and sixth-order SD schemes [128].
Fig. 31. Lines formed around the NACA0012 airfoil at M=0.5, Re=5000, α=0
(courtesy of Fidkowski, Darmofal et al. [146]).
Fig. 32. Element line smoother convergence histories vs. grid size for inviscid duct flow at M=0.5 (courtesy of Fidkowski et al. [146]).
Fig. 33. A typical two level h-multigrid mesh configuration (courtesy of Nastase and Mavriplis [141]).
Fig. 34. The full hp-multigrid convergence vs. the number of multigrid (MG) cycles, on a mesh size of N=5916 elements and polynomial degrees (p), for the four-element airfoil problem (courtesy of Nastase and Mavriplis [41]).
Fig. 35. The full hp-multigrid convergence vs. the number of multigrid (MG) cycles, on various fine grid problem sizes and polynomial degree 4, for the four-element airfoil problem (courtesy of Nastase and Mavriplis [41]).
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