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Spectral (Finite) Volume Method for One Dimensional Euler Equations 

Z.J. Wang (zjw@egr.msu.edu) Yen Liu (liu@nas.nasa.pov) 

„ ^ i- » , u ■ 1 T- Research Scientist 

Associate Professor of Mechanical Engineenng ^t a o * a t> i, r- «.». 

NASA Ames Research Center 
Michigan State University, 2555 EB, j^^^^^^ Pj^j^ ^^ 94035 y g^ 

East Lansing, MI 48824, U.S.A. 

1. Basic Idea 

Consider a mesh of unstructured triangular cells. Each cell is called a Spectral Volume (SV), denoted by S„ which is 
further partitioned into subcells named Control Volumes (CVs), indicated by Qy, as shown in Figure la. To 
represent the solution as a polynomial of degree m in two dimensions (2D) we need N = (m+l)(m+2)r2 pieces of 
independent information, or degrees of freedom (DOFs). The DOFs in a SV method are the volume-averaged mean 
variables at the N CVs. For example, to build a quadratic reconstruction in 2D, we need at least (2+l)(3+l)/2 = 6 
DOFs. There are numerous ways of partitioning a SV, and not every partition is admissible in the sense that the 
partition may not be capable of producing a degree m polynomial. Once A' mean solutions in the CVs of a SV are 
given, a unique polynomial reconstruction can be obtained from 

N 

where p^{x,y)e P„ (the space of polynomials of degree m or less), Lj(x.y)E P„J = l---,N are the "shape" 
functions satisfying 

JL,ix,y)dV=\c,j\S^,. (2) 

This high-order polynomial reconstruction facilitates a high-order update for the mean solution of each CV. 
Consider the following hyperbolic conservation law 

«, + V • F = , (3) 

where F is called the flux vector. A high-order update can be obtained by integrating (3) in each CV to obtain 

— ic,,i+y i'(^««)^=o, (4) 

dt I '■-'I ^J 

'■=1 A, 

where K is the total number of faces in C,j, and u,j is the volume-averaged solution at C,j. The flux integral in (4) 

is then replaced by an m* order Gauss-quadrature formula 

J 
|(F./i)dA = ^Hv,F(M(x,,,y,,)).n,A„ (5) 

where J is the number of quadrature points on the r-th face, w,, are the Gauss quadrature weights, (Xr^, >v<,) are the 
Gauss quadrature points. Since the reconstructed polynomials are piece-wise continuous, the solution is usually 
discontinuous across the boundaries of a SV. The fluxes at the interior faces can be computed directly based on the 
reconstructed solution at the quadrature points. The fluxes at the boundary faces of a SV are computed using 
approximate Riemann solvers given the left and right reconstructed solutions, i.e., 

FiuiXrg , yrq )) » « = pRiem iPiiXrg , J rq ), PiA^rg , }' rq )• "r \ ^^^ 

where p,,(x, ,y, ) is the reconstructed polynomial in a neighboring SV sharing face r with the SV in 
consideration, 5,. It has been shown [1] that resultant order of accuracy of this SV scheme is (m+l)-th order. In 
addition, the scheme is compact in the sense that a high-order polynomial is reconstructed in each SV without using 
any data from neighboring SVs. This property can potentially translate into significantly reduced communication 
cost compared to a high-order k-exact finite volume (FV) method [2] when implemented on parallel computers. In 
addition, the SV method is much more efficient than the k-exact method because its reconstruction can be solved 
analytically, and the reconstruction is universal for the same partition, irrespective of spectral volume shapes. The 
reconstruction problem in a k-exact method must be solved numerically for each cell because the reconstruction 
stencil is different for each cell, as shown in Figure lb. 



The SV method is similar to the Discontinuous Galerkin (DG) [3] method in philosophy. However, the SV method 
has the following advantages compared to the DG method: 1) better stability limit ; 2) unknowns updated 
independently and no mass matrix inversion; 3) only m'h order surface integration required for (m+l)* order 
schemes, not 2mth order surface and volume integrations as in the DG method; 4) higher-resolution for 
discontinuities because limiters can be designed for the CV means rather than the SV means; 5) the avoidance of a 
high-order volume integral, which is required in the DG method. 

2. Some Results 

Through simple analysis and numerical tests, it turns out the accuracy, stability and convergence property of the SV 
method hinges on the proper partition of a SV into CVs. In one-dimensional tests, it has been shown that a uniform 
partition of a SV is not convergent for higher than fourth-order SV schemes. A partition using the Gauss-Lobatto 
points has given excellent results. An accuracy study has confirmed the expected high-order accuracy with the SV 
method. 

The SV method has been successfully demonstrate for both ID and 2D scalar conser\'ation laws [1]. In this paper, 
the SV method is further extended to the one-dimensional system - the Euler equations. Roe's Riemann solver is 
employed to compute the flux across SV boundary faces. One demonstration case of blast wave interaction is shown 
in Figure 2a, which presents the computed density profile with a third-order SV method. The solid line represents a 
"converged" solution using a MUSCL scheme on a very fine mesh. The computational density profile with a third- 
order DG scheme is shown in Figure 2b. Note that the SV scheme has far better resolution than the DG scheme. 





(a) (b) 

Figure 1. (a) The partition of a Spectral Volume into six Control Volumes supporting quadratic reconstruction; (b) A 
possible reconstruction stencil for a quadratic reconstruction in a high-order finite volume scheme; 




(a) (b) 

Figure 2. Third-Order SV Method (a) and DG Method (b) with TVD Limiters, 400 Cells 
References 

1. Z.J. Wang and Yen Liu, "Finite spectral volume method for conservation laws on unstructured grids: 2D scalar 
equations," submitted to J. Comput. Phys. 

2. T.J. Barth and P.O. Frederickson, "High-order solution of the Euler equations on unstructured grids using 
quadratic reconstruction," AIAA Paper No. 90-0013, 1990. 

3. B. Cockbum, S.-Y. Lin and C.-W. Shu, "TVB Runge-Kutta local projection discontinuous Galerkin finite 
element method for conservation laws III: one-dimensional systems," J. Comput. Phys. 84, 90-113 (1989).