         !" #$%  &       ’&   & ( %)) %%*$ +   ,$ ’%* -* .+  %*- /$ %!+ 0 *Vrije Universiteit Brussel, Department of Mechanical Engineering, Fluid Dynamics and Thermodynamics Research Group Triomflaan 43, B$1050 Brussels, Belgium 1 Email: mparsani@vub.ac.be, web page: http://mech.vub.ac.be/thermodynamics 2 Email:kvdabeel@vub.ac.be, web page: http://mech.vub.ac.be/thermodynamics 3 Email:chris.lacor@vub.ac.be, web page: http://mech.vub.ac.be/thermodynamics ,1 "-2 Spectral Volume Method, LUSGS Algorithm, pMultigrid, Residual Restriction Operator. )%!)3 An efficient implicit lowerupper symmetric GaussSeidel (LUSGS) solution algorithm has been combined for the first time with a highorder spectral volume method and a full p$Multigrid strategy to accelerate the convergence rate. The LUSGS solver is preconditioned by the block element matrix, and the system of equations is the solved with an exact LU decomposition approach with pivoting per columns. The time integration methods considered are the backward Euler difference, the second and thirdorder backwards differencing formulae and the fourthorder explicitfirststage, singlediagonal coefficient, diagonallyimplicit RungeKutta method. The implicit solver with backwards schemes has shown a speed up factor of more than one order of magnitude relative to the wellknown multistage optimized RungeKutta schemes for the quasi1D Euler flow while the explicitfirststage, single diagonalcoefficient, diagonallyimplicit RungeKutta method has demonstrated to be around four times more efficient then the optimized explicit RungeKutta smoothers. In addition, the efficiency of the p$ Multigrid algorithm is investigated with two different residual restriction operators. The first one was proposed by K. Van den Abeele et al. 21 and it is welldefined for certain 1D problems, while the second one is presented in this paper and it is more general because it treats the residual which arise from the spatial discretization as a controlvolume average quantity. It has shown that for the quasi1D Euler flow and the optimized explicit RungeKutta smoothers the residual restriction operator proposed by K. Van den Abeele et al. 21 is more efficient than the general one. On the other hand, the two residual restriction operators do not show any differences in the convergence rates of the implicit LUSGS solver. 3 & Highorder spatial accurate numerical schemes are being developed for use in a variety of solution procedures. In computational fluid dynamics (CFD), highorder accurate schemes are being pursued for direct numerical, large eddy, computational aeroacoustic, turbulent combustion and biomedics simulations where accurate resolution of small scales requires large grid densities. In addition, since CFD is more and more used as an industrial design and analysis tool, the applications are often of industrial relevance,