The compact high-order "Spectral Volume Method" designed for conservation laws on unstructured grids is presented. Its spectral reconstruction is exposed briefly and its applications to the Euler equations are presented through several test cases to assess its accuracy and stability. Comparisons with classical methods such as MUSCL show the superiority of SVM. The SVM method arises as a high-order accurate scheme, geometrically flexible and computationally efficient

Chikhaoui, Oussama

Gressier, Jérémie

Grondin, Gilles

English

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  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 4362 To cite this document: CHIKHAOUI Oussama, GRESSIER Jérémie, GRONDIN Gilles. Assessment of the Spectral Volume Method on inviscid and viscous flows. In: The Sixth International Conference on Computational Fluid Dynamics - ICCFD, 12-16 Jul 2010, St-Pertersburg, Russia.   Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr Assessment of the Spectral Volume Method onInviscid and Viscous FlowsOussama Chikhaoui, Je´re´mie Gressier and Gilles GrondinAbstract The compact high-order ’Spectral Volume Method’ (SVM, Wang (2002))designed for conservation laws on unstructured grids is presented. Its spectral recon-struction is exposed briefly and its applications to the Euler equations are presentedthrough several test cases to assess its accuracy and stability. Comparisons withclassical methods such as MUSCL show the superiority of SVM. The SVM methodarises as a high-order accurate scheme, geometrically flexible and computationallyefficient.1 INTRODUCTIONDespite the constant improvements in computational and data processing resources,the continuously growing requirements of computational fluid dynamics still re-main unsatisfied. In the last decade, the CFD community showed a growing inter-est in high-order approximations to address these issues (WENO, DiscontinuousGalerkin, . . . ). An attractive choice is the ’Spectral Volume Method’ (SVM) pro-posed and developed by Wang et al. [6, 5, 3] which achieves high-order accuracy onunstructured grids through polynomial reconstruction.To assess the performance of the SVM method, different test cases were com-puted with Typhon, an unstructured open-source code. The numerical experimentswere chosen to cover a large set of flow configurations from continuous quasi-incompressible problems to shock wave propagations and mixing flows.The results presented here are up to expectations with a significant increase inaccuracy and a reduction in CPU time compared to a usual second order method.De´partement d’Ae´rodynamique, E´nerge´tique et PropulsionInstitut Supe´rieur de l’Ae´ronautique et de l’Espace10 av. Edouard Belin - BP 54032 - 31055, Toulouse, FRANCEe-mail: {oussama.chikhaoui, jeremie.gressier, gilles.grondin}@isae.fr122 Spectral Volume Method for the 2D Euler equationsThe SVM method achieves high-order accuracy on unstructured grids through poly-nomial reconstruction within initial grid cells (spectral volumes, SVs) with a subdi-vision of the each SV into polygonal control volumes (CVs, Fig. (1)). The spectralsplitting of the SV is designed to minimize internal reconstruction oscillations [1].The flux computations are finally achieved with Gaussian quadrature directly in-ferred from CV states ponderation with a constant and unique set of coefficients foreach SV in the whole domain.A BCDEFGA BCGD EILJMF HKA1A BCD E FIPTJRUG HSONKLMQFig. 1 Geometric splitting definitions for the second, third and fourth order partitionsWhile high-order reconstructions are often based on projections and gradientevaluations on a quite large region of neighboring cells, the SVM compact sten-cil makes it computationally efficient and attractive. The polynomial reconstructionused remains exact (at a given order) on arbitrarily shaped triangles.2.1 Spectral volumes partitionsThe subdivision of a spectral volume Si in different controle volumes Ci j is an es-sential step for the SVM scheme. For example, some geometric partitions can leadto degenerated systems and thus should be excluded. The splitting procedure mustconserve the existing symmetries in a triangle, use straight edges and consider con-vex CVs. Fig. 1 shows different geometric partitions of a spectral volume for differ-ent desired orders. While the second order splitting definition is unique, the higherorder partitions introduce some geometric parameters. The third order partitions isdefined by two parameters: α = ADAB and β =AFAA1 . The fourth order partitions havefour degrees of freedom: α = ADAB , β =AGAR , γ =ORAR and δ =ALAR . The accuracy andstability of a given spectral volume scheme depends on the choice of these geomet-ric parameters [1, 2].Table 1 sums up the different implemented and tested partitions with their geo-metric parameters.3Partition Order α β δ γSVM2 2 - - - -SVM3W 3 1/4 1/3 - -SVM3K 3 91/1000 18/100 - -SVM3K2 3 0.1093621117 0.1730022492 - -SVM4W 4 1/15 2/15 2/15 1/15SVM4K 4 78/1000 104/1000 52/1000 351/1000SVM4K2 4 0.0326228301 0.042508082 0.0504398911 0.1562524902Table 1 Geometric parameters for the different SVM partitionsThe SVMW splittings were proposed by Wang et al. and designed by minimizingthe Lebesgue constant over the SV. Whereas the minimization of this constant pro-vides a good assessment of the quality of the SVM splitting, different observationsshow that it is not a sufficient condition for the stability of the scheme. Thus, Abeeleproposed other geometric splittings noted as SVMK and SVMK2 [1, 2] .2.2 Spectral volume method assetsThe spectral volume method uses a compact stencil which is a great advantage com-pared to other high order reconstructions. The SVM geometric splitting is designedand optimised in a spectral way to minimize internal reconstruction oscillationsknown as Runge phenomena. The reconstructed field is continuous over the entireSV, therefore internal faces do not constitute Riemann problems, which reduces theflux computation cost and contains the problem of data limitation to the SV faces.The other interesting aspect of the SVM is the homothetic nature of the splittingreconstruction: no new metric terms need be kept in memory. The interpolation onGauss points for fluxes computation is directly inferred from a weighting of CVstates with constant and unique coefficients for the whole domain.Lastly, while the usual finite-volume and finite-difference methods depend stronglyon the grid quality and density, the SVM reconstruction remains exact (at a givenorder) on arbitrarily shaped triangles.3 Numerical experimentsTo assess the performance of the SVM method, different test cases were computedwith Typhon, an open-source unstructured solver [4]. The numerical experimentswere chosen to cover a large set of flow configurations. In the next sections, we focuson the results of two chosen cases: the evolution of a convected vortex and a simpleMach reflection on a wedge. The HLLC solver is used for the Riemann problem atfaces. Time integration is performed with a third order TVD Runge-Kutta schemeand SVM results are compared to results provided by a usual MUSCL method.43.1 Convected VortexWe consider a vortex evolution problem governed by: ∂ p∂ t = ρvθ 2r and convected witha speed of Vconv = 20 in the x direction on a regular domain of [−5 : 5]× [−5 : 5].Periodic boundary conditions are set in the x and y directions. Thus, the vortexcrosses the whole domain twice from left to right between t=0 and t=1. No limiterswere employed for the SVM simulations.Fig. 2 Overview of numerical damping of a convected vortex using MUSCL schemeThe overview of the vortex damping is presented on figure 2 for the MUSCLsimulation using the Van Albada limiter. While the theoretical maximum transversevelocity is Vymax = 30, Vymax ∼ 10 is obtained for this MUSCL simulation, therebyshowing that this test case is very sensitive to numerical dissipation.For comparison purposes, the velocity profiles are considered on figure 3 at t=1along a horizontal line passing through the vortex center. With the SVM schemes,the velocity profile is better preserved when the order is increased. Thus, Vymax ∼15, Vymax ∼ 25 and Vymax ∼ 30 are obtained for the second, third and fourth orderrespectively.Another interesting aspect of SVM simulations is the reduction in CPU timeper time step compared to a classic MUSCL method using the same number ofcontrol volumes: 50% for 2nd order, 32% to 35% for 3rd order and 22% to 25% for4th order. The CPU time savings are due to the absence of gradient evaluation forinviscid fluxes and the continuity of state variables through internal faces. This latterfact reduces the limitation problems and several cases can be computed without anylimitation procedure.5Fig. 3 Comparison of velocity profiles along a line through the vortex center at t = 1 (left) andkinetic energy time evolution (right)3.2 Simple Mach reflectionThis case deals with a classic problem of shock reflection with a Mach numberMs = 1.7 and a wedge angle of θ = 25◦. The numerical results were obtained on adomain of [25×16.5] on the x,y plane with the apex of the wedge placed at x,y =4.69,0. The upstream shock conditions are ambient conditions, with ρa = 1.225,pa = 1.01325 ·105 and u= 0.Different SVM simulations were carried out with different orders. Observationsshow that the contours around the incident and reflected shocks and the slip sur-face are better captured when the order increases. To prove that the capture of thesefeatures and the better resolution of the flow are due to the increasing order, othersimulations have been performed using the MUSCL scheme. The results obtainedFig. 4 Entropy contours at slip surface: MUSCL, 800.000 CV (left) and SVM4, 422.080 CV (right)with the SVM4 scheme were compared to those provided by the MUSCL methodon a finer structured cartesian grid. While the unstructured grid for SVM4 contains422.080 CVs, the structured grid used for MUSCL simulation contains almost twice6the number of CVs, i.e. 800.000 CVs. Yet, the shocks thicknesses obtained are largeand no improvement of the slip surface resolution is observed when increasing thenumber of CVs with the MUSCL method. Figure 4 clearly shows that the SVMscheme better captures the slip surface: Kelvin-Helmholtz instabilities can be ob-served on the shear line. These observations suggest that the better resolution of theproblem is achieved by the order increase rather than the mesh refinement.4 ConclusionIn this study, analysis and applications of the Spectral Volume Method through ap-plications were presented. This high-order reconstruction is particularly interestingowing to its compact support. The use of flux evaluation on split control volumesmakes this method very similar to Finite-Volume method which makes the SVMimplementation in existing codes relatively easy.The results are up to expectations with a significant increase in occuracy and areduction in CPU time compared to a MUSCL method with the same number ofelements. The CPU gains are due to the absence of gradient evaluation for inviscidfluxes and the continuity of status variables through internal faces. This property re-duces the limitation problems and several cases can be computed with no limitationprocedure.For all previously reported assets, the Spectral Volume Method arises as apromising high-order reconstruction mode for both academic and industrial stud-ies especially for complex applications which require great accuracy with computa-tional resource savings.References1. Abeele, K. and Lacor., C. An accuracy and stability study of the 2d spectral volume method.Journal of Computational Physics, 226(2):1007–10026, May 2007.2. Abeele, K., Ghorbaniasl, M., Parsani, M., Lacor C., A stability analysis for the spectralvolume method on tetrahedral grids Journal of Computational Physics, 228, pp. 257-265,2009.3. Sun, Y. , Wang, Z.J., and Liu. Y., Spectral (finite) volume method for conservation laws on un-structured grids vi: Extension to viscous flow. Journal of Computational Physics, 215(1):41–58, June 2006.4. Typhon, http://typhon.sourceforge.net/5. Wang., Z. J., Spectral (finite) volume method for conservation laws on unstructured grids.basic formulation: Basic formulation. Journal of Computational Physics, 178(1):210–251,May 2002.6. Wang, Z. J., Zhang, L., and Liu., Y. Spectral (finite) volume method for conservation lawson unstructured grids iv: extension to two-dimensional systems. Journal of ComputationalPhysics, 194(2):716–741, March 2004.

A stability analysis for the spectral volume method on tetrahedral grids

An accuracy and stability study of the 2d spectral volume method.

Spectral (ﬁnite) volume method for conservation laws on unstructured grids iv: extension to two-dimensional systems.

Spectral (ﬁnite) volume method for conservation laws on unstructured grids vi: Extension to viscous ﬂow.

Spectral (ﬁnite) volume method for conservation laws on unstructured grids. basic formulation: Basic formulation.

Assessment of the Spectral Volume Method on inviscid and viscous flows

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Assessment of the Spectral Volume Method on inviscid and viscous flows

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