A spectral volume Navier-Stokes solver on unstructured tetrahedral grids

The spectral volume (SV) method was first introduced as a method to solve systems of convection equations, like the Euler equations, in a series of papers by Wang et al., e.g. [1] and the references therein. It is related to the discontinuous Galerkin (DG) method, in the sense that it also uses high order polynomials to approximate the solution in each grid cell, and Riemann solvers to deal with the discontinuities in the solution along the cell faces. The SV method can be extended to systems of convection diffusion equations, like the Navier Stokes (NS) equations, in a similar manner as the DG method, see [2]. Further contributions to the development of the SV method were made in Van den Abeele et al. [3,4], where the method stability was analyzed and stable schemes were derived for 1D and for 2D triangular grids.
In the present contribution, an implementation in the COOLFluiD code [5], which was developed at the Von Karman Institute, of the SV method for the N-S equations on tetrahedral grids is presented. The implementation features both the quadrature approach, where Gaussian quadrature formulas are used for the evaluation of the residual integrals, and the quadrature free approach, where such formulas are avoided. The latter approach was recently described for the 2D Euler equations in Harris et al. [6]. In the present implementation, for an order of acccuracy higher than two, a significant decrease in computational effort for the evaluation of the residuals was observed with the quadrature free approach, while the order of accuracy was maintained. For the discretization of the diffusive terms, an approach similar to the local approach for the DG method, as described in Cockburn and Shu [7], was followed. As an example, the mach contours for the flow around a NACA0012 airfoil at Re = 5000 and M = 0.5,
obtained with a third order SV scheme, is shown in the left plot of Figure 1.
While high order accurate compact schemes, such as the SV and the DG methods, can yield accurate results more quickly than traditional low order schemes, fast and robust solvers are a necessity to fullfill this potential. This is illustrated in Figure 2, where the Fourier footprints (FFs) corresponding to second, third and fourth order accurate SV schemes for the 1D linear diffusion equation du/dt = d2u/dx2 are plotted. These FFs were computed in an analogous way as described in [3]. It is obvious from the plots that the size of the FF increases dramatically with the polynomial order of the SV schemes. Consequently, there is a severe restriction on the maximum time step that preserves stability, if traditional explicit Runge Kutta (RK) schemes, as described in [3], are used as solvers for these schemes. Therefore, an implicit solver based on the backward Euler scheme was implemented. This scheme leads to a system of nonlinear equations at every iteration, which is linearized, and the resulting system of linear
equations is inverted using a generalized minimal residual method. The histories of the mass density residual obtained with an explicit five stage RK solver and the implicit solver are shown in the middle and right plots of Figure 1. Clearly, the implicit solver needs far less iterations and is able to converge the solution much further than the explicit one. Moreover, the implicit solver is much more efficient in terms of CPUtime.

REFERENCES
[1] Z. J. Wang, L. Zhang, and Y. Liu, Spectral (finite) volume method for conservation laws on
unstr. grids IV: Ext. to 2D Euler equations. J. Comput. Phys., Vol. 194(2), 716 741, 2004.
[2] Y. Sun, Z.J. Wang and Y. Liu, Spectral (finite) volume method for conservation laws on
unstructured grids VI: Extension to viscous flow. J. Comput. Phys., Vol. 215, 41 58, 2006.
[3] K. Van den Abeele, T. Broeckhoven and C. Lacor, Disp. and diss. prop. of the 1D spectral
vol. method and appl. to a p multigr. alg. . J. Comput. Phys., Vol. 224(2), 616 636, 2007.
[4] K. Van den Abeele and C. Lacor, An accuracy and stability study of the 2D spectral volume
method. J. Comput. Phys., Vol. 226(1), 1007 1026, 2007.
[5] http://coolfluidsrv.vki.ac.be/coolfluid/
[6] R. Harris, Z.J. Wang and Y. Liu, Efficient quadrature free high order spectral volume
method on unstructured grids: Theory and 2D implementation. J. Comput. Phys., 2007,
doi: 10.1016/j.jcp.2007.09.012.
[7] B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time dependent
convection diffusion systems. SIAM J. Numer. Anal., Vol. 35, 2440 2463, 1998.

Publication year:2008

Keywords:spectral volume method, Navier-Stokes equations, implicit solution algorithms