Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-17T13:07:10.790Z Has data issue: false hasContentIssue false
  • English
  • Français

An Optimized Correction Procedure via Reconstruction Formulation for Broadband Wave Computation

Published online by Cambridge University Press:  03 June 2015

Yi Li  and
Z.J. Wang
Yi Li*
Affiliation:
Department of Aerospace Engineering and CFD Center, Iowa State University, Ames, IA 50011, USA
Z.J. Wang*
Affiliation:
Department of Aerospace Engineering and CFD Center, Iowa State University, Ames, IA 50011, USA
*
Corresponding author.Email:yili@iastate.edu
Email:zjw@iastate.edu
Get access

Abstract

Recently, a new differential discontinuous formulation for conservation laws named the Correction Procedure via Reconstruction (CPR) is developed, which is in-spired by several other discontinuous methods such as the discontinuous Galerkin (DG), the spectral volume (SV)/spectral difference (SD) methods. All of them can be unified under the CPR formulation, which is relatively simple to implement due to its finite-difference-like framework. In this paper, a different discontinuous solution space including both polynomial and Fourier basis functions on each element is employed to compute broad-band waves. Free-parameters introduced in the Fourier bases are optimized to minimize both dispersion and dissipation errors through a wave propagation analysis. The optimization procedure is verified with a mesh resolution analysis. Numerical results are presented to demonstrate the performance of the optimized CPR formulation.

Keywords

76CPR (correction procedure via reconstruction)hybrid discontinuous spacewave propagation analysisunstructured meshes
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013 , pp. 1265 - 1291
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abgrall, R., Roe, P.L., High order fluctuation schemes on triangular meshes, J. Sci. Comput. 19 (2003) 3–36.Google Scholar
[2]Ashcroft, G., Zhang, X., Optimized prefactored compact schemes, J. Comput. Phys. 190 (2003) 459–477.Google Scholar
[3]Balsara, D., Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasing high order of accuracy, J. Comput. Phys. 160 (2000) 405–452.Google Scholar
[4]Barth, T.J., Frederickso, P.O., High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper 96–0027,1996.Google Scholar
[5]Bassi, F., Rebay, S., High-order accurate discontinuous finite element solution of the 2D euler equations, J. Comput. Phys. 138 (1997) 251–285.Google Scholar
[6]Christofi, S., The study of building blocks for essentially non-oscillatory (ENO) schemes, Ph.D. thesis, Division of Applied Mathematics, Brown University, 1996.Google Scholar
[7]Cockburn, B., Li, F., Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys. 194 (2004) 588–610.Google Scholar
[8]Cockburn, B., Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comput. 52 (1989) 411–435.Google Scholar
[9]Cockburn, B., Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys. 141 (1998) 199–224.CrossRefGoogle Scholar
[10]Costa, B., Don, W.-S., Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws, J. Comput. Phys. 224 (2007) 970–991.Google Scholar
[11]Ekaterinaris, J.A., High-order accurate, low numerical diffusion methods for aerodynamics, Prog. Aerosp. Sci. 41 (2005) 192–300.CrossRefGoogle Scholar
[12]Gao, H., Wang, Z.J., A conservative correction procedures via reconstruction formulation with the Chain-rule divergence evaluation, submitted.Google Scholar
[13]Hixon, R., A new class of compact schemes, AIAA paper 98–0367,1998.Google Scholar
[14]Hu, C.-Q., Shu, C.-W., Weighted essentially Non-oscillatory schemes on Triangular meshes, J. Comput. Phys. 150 (1999) 97–127.Google Scholar
[15]Hu, F.-Q., Hussaini, M.Y., Rasetarinera, P., An analysis of the discontinuous Galerkin method for wave propagation problems, J. Comput. Phys. 151 (1999) 921–946.Google Scholar
[16]Huang, P.G., Wang, Z.J., Liu, Y., An implicit space-time spectral difference method for discontinuity capturing using adaptive polynomials, AIAA Paper 2005–5255, 2005.Google Scholar
[17]Huynh, H.T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA paper 2007–4079, 2007.Google Scholar
[18]Jurgens, H.-M., High-accuracy finite-difference schemes for linear wave propagation, Ph.D thesis, Department of Aerospace Science and Engineering, University of Toronto, 1997.Google Scholar
[19]Kadalbajoo, M.K., Patidar, K.C., Exponentially fitted spline in compression for the numerical solution of singular perturbation problems, Comput. Math. Appl. 46 (2003) 751–767.Google Scholar
[20]Kopriva, D.-A., Kolias, J.-H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comput. Phys. 125 (1996) 244–261.Google Scholar
[21]Kopriva, D.-A., A staggered-grid multidomain spectral method for compressible Navier-Stokes equations, J. Comput. Phys. 143 (1998) 125–158.Google Scholar
[22]Li, F., Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for MHD equations, J. Sci. Comput. 22-23 (2005) 413–442.Google Scholar
[23]Li, F., Shu, C.-W., Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Math. Lett. 18 (2005) 1204–1209.Google Scholar
[24]Li, F., Shu, C.-W., A local-structure-preserving local discontinuous Galerkin method for the Laplace equation, Meth. Appl. Anal. 13 (2006) 215–234.Google Scholar
[25]Li, Y.-G., Wavenumber-extended high-order upwind-biased finite-difference schemes for convective scalar transport, J. Comput. Phys. 133 (1997) 235–255.Google Scholar
[26]Liu, Y., Vinokur, M., Wang, Z.J., Discontinuous spectral difference methods for conservation laws on unstructured grids, in Groth, C. and Zingg, D.W. (Eds.), Proceeding of the 3rd international conference in CFD, Toronto, Springer, 2004, pp. 449–454.Google Scholar
[27]Liu, Y., Vinokur, M., Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems, J. Comput. Phys. 212 (2006) 454–472.CrossRefGoogle Scholar
[28]Liu, Y., Vinokur, M., Wang, Z.J., Spectral difference methods for unstructured grids I. Basic formulation, J. Comput Phys. 216 (2006) 780–801.CrossRefGoogle Scholar
[29]May, G., Jameson, A., A spectral difference method for the Euler and Navier-Stokes equations on unstructured grids, AIAA Paper 2006-304, 2006.Google Scholar
[30]Patera, A.T., A spectral element method for fluid dynamics: Laminar flow in a channel ex-pansion, J. Comput. Phys. 54 (1984) 468–488.Google Scholar
[31]Reddy, Y.N., Chakravarthy, P.P., An exponentially fitted finite difference method for singular perturbation problems, Appl. Math. Comput. 154 (2004) 83.Google Scholar
[32]Sia, H.-Q., Wang, T.-G., Grid-optimized upwind dispersion-relation-preserving scheme on non-uniform Cartesian grids for computational aeroacoustics, Aerosp. Sci. Tech. 12 (2008) 608–617.Google Scholar
[33]Sun, Y., Wang, Z.J., Liu, Y., Spectral (finite) volume method for conservation laws on unstruc-tured grids VI: Extension to viscous flow, J. Comput. Phys. 215 (2006) 41–58.Google Scholar
[34]Sun, Y., Wang, Z.J., Liu, Y., High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids, Commun. Comput. Phys. 2(2) (2007) 310–333.Google Scholar
[35]Sun, Y., Wang, Z.J., Liu, Y., Chen, C.L., Efficient implicit LU-SGS algorithm for high-order spectral difference method on unstructured hexahedral grids, AIAA Paper, 2007–0313,2007.Google Scholar
[36]Tam, C.K.W., Computational aeroacoustics: Issues and methods, AIAA J., 33(10) (1995) 1788–1796.CrossRefGoogle Scholar
[37]Tam, C.K.W., Webb, J.-C., Dispersion-relation-preserving finite difference schemes for com-putational acoustics, J. Comput. Phys. 107 (1993) 262–281.Google Scholar
[38]Van den Abeele, K., Development of high-order accurate schemes for unstructured grids, Ph.D thesis, Department of Mechanical Engineering, Vrije Universiteit Brussel, Belgium, 2008.Google Scholar
[39]Van den Abeele, K., Lacor, C., An accuracy and stability study of the 2D spectral volume method, J. Comput. Phys. 226 (2007) 1007–1026.Google Scholar
[40]Van den Abeele, K., Lacor, C., Wang, Z.J., On the connection between the spectral volume and the spectral difference method, J. Comput. Phys. 227 (2007) 877–885.Google Scholar
[41]Van den Abeele, K., Lacor, C., Wang, Z.J., On the stability and the accuracy of the spectral difference method, J. Sci. Comput. 37(2) (2008) 162–188.Google Scholar
[42]Van den Abeele, K., Broeckhoven, T., Lacor, C., Dispersion and dissipation properties of the 1D spectral volume method and application to a p-multigrid algorithm, J. Comput. Phys. 224 (2007) 616–636.Google Scholar
[43]Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys. 178 (2002) 210–251.Google Scholar
[44]Wang, Z.J., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, J. Prog. Aerosp. Sci. 43 (2007) 1–41.Google Scholar
[45]Wang, Z.J., Chen, R.F., Optimized weighted essentially nonoscillatory schemes for linear waves with discontinuity, J. Comput. Phys. 174 (2001) 381–404.Google Scholar
[46]Wang, Z.J., Gao, H.-Y, A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys. 228 (2009) 8161–8186.Google Scholar
[47]Wang, Z.J., Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids II: Extension to two-dimensional scalar equation, J. Comput. Phys. 179 (2002) 665–697.Google Scholar
[48]Wang, Z.J., Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids III: Extension to one-dimensional systems, J. Sci. Comput. 20 (2004) 137–157.Google Scholar
[49]Wang, Z.J., Zhang, L., Liu, Y., Spectral (finite) volume method for conservation laws on un-structured grids III: Extension to two-dimensional systems, J. Sci. Comput. 194 (2004) 716–741.Google Scholar
[50]Wang, Z.J., Liu, Y., The spectral difference method for the 2D Euler equations on unstructured grids, AIAA Paper 2005–5112,2005.Google Scholar
[51]Wang, Z.J., Liu, Y., May, G., Jameson, A., Spectral difference method for unstructured grids II: Extension to the Euler equations, J. Sci. Comput. 32(1) (2007) 45–71.CrossRefGoogle Scholar
[52]Yuan, L., Shu, C.-W., Discontinuous Galerkin method based on non-polynomial approximation spaces, J. Comput. Phys. 218 (2006) 295–323.Google Scholar
[53]Zingg, D.W., A review of high-order and optimized finite difference methods for simulating linear wave phenomena, AIAA Paper 97-2088,1997.Google Scholar
[54]Zhuang, M., Chen, R.F., Optimized upwind dispersion-relation-preserving finite difference scheme for computational aeroacoustics, AIAA J. 36(11) (1998) 2146–2148.Google Scholar
[55]Zhuang, M., Chen, R.F., Application of high-order optimized upwind schemes for computational aeroacoustics, AIAA J. 40(3) (2002) 443–449.Google Scholar