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Analysis of Spectral Volume Methods for 1D Linear Scalar Hyperbolic Equations

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Abstract

This paper is concerned with the analysis of two spectral volume (SV) methods for 1D scalar hyperbolic equations : one is constructed basing on the Gauss–Legendre points (LSV) and the other is based on the right-Radau points (RRSV). We first prove that for a general nonuniform mesh and any polynomial degree k, both the LSV and RRSV methods are stable and can achieve optimal convergence orders in the \(L^2\) norm. Secondly, we prove that both methods have some superconvergence properties at some special points. For instances, at the downwind points, the solution of RRSV and LSV methods converges with the order of \(\mathcal{O}(h^{2k+1})\) and \(\mathcal{O}(h^{2k})\), respectively. Moreover, we demonstrate that for constant-coefficient equations, the RRSV method is identical to the upwind discontinuous Galerkin (DG) method. Our theoretical findings are validated with several numerical experiments at the end.

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References

  1. Adjerid, S., Massey, T.C.: Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput. Methods Appl. Mech. Eng. 195, 3331–3346 (2006)

    Article  MathSciNet  Google Scholar 

  2. Adjerid, S., Weinhart, T.: Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems. Math. Comput. 80, 1335–1367 (2011)

    Article  MathSciNet  Google Scholar 

  3. Barth,T., Ohlberger,M.: Finite volume methods: foundation and analysis. In: Encyclopedia of Computational Mechanics, vol. 1, pp. 439–474. John Wiley & Sons, Chichester (2004)

  4. Cao, W., Zhang, Z., Zou, Q.: Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 52, 2555–2573 (2014)

    Article  MathSciNet  Google Scholar 

  5. Cao, W., Shu, C.-W., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients. ESIAM: M2AN 51, 2213–2235 (2017)

    Article  MathSciNet  Google Scholar 

  6. Cao, W., Shu, C.-W., Yang, Y., Zhang, Z.: Superconvergence of Discontinuous Galerkin method for nonlinear hyperbolic equations. SIAM J. Numer. Anal. 56, 732–765 (2018)

    Article  MathSciNet  Google Scholar 

  7. Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044–4072 (2010)

    Article  MathSciNet  Google Scholar 

  8. Choi, B., Iskandarani, M., Levin, J., Haidvogel, D.: A spectral finite-volume method for the shallow water equations. Mon. Weather Rev. 132, 1777–1791 (2004)

    Article  Google Scholar 

  9. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws, II: general framework. Math. Comput. 52, 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Cockburn, B., Dong, B., Guzman, J.: Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46, 1250–1265 (2008)

    Article  MathSciNet  Google Scholar 

  11. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws, V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MathSciNet  Google Scholar 

  12. Cockburn, B., Karniadakis, G., Shu, C.-W.: The Development of Discontinuous Galerkin Methods. Springer, Berlin (2000)

    Book  Google Scholar 

  13. Cozzolino, L., Della Morte, R., Del Giudice, G.: A well-balanced spectral volume scheme with the wetting drying property for the shallow-water equations. J. Hydroinform 14, 745–760 (2012)

    Article  Google Scholar 

  14. Despres,B., Kokh,S., Lagoutiere,F.: Sharping methods for finite volume schemes. In: Handbook of Numerical Analysis, vol. XVII, pp. 77–102 (2016)

  15. Eymard, R., Gallouet, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)

  16. Haga, T., Furndate, M., Sawada, K.: RANS simulation using high-order spectral volume method on unstructured tetrahedral grids. In: AIAA Paper 2009-404 (2009)

  17. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)

    Article  MathSciNet  Google Scholar 

  18. Harris, R., Wang, Z.: High-order adaptive quadrature-free spectral volume method on unstructured grids. Comput. Fluids 38, 2006–2025 (2009)

    Article  Google Scholar 

  19. Harris, R., Wang, Z.: Partition design and optimization for high-order spectral volume schemes. In: AIAA Paper 2009-1333 (2009)

  20. Houston, P., Schwab, C., Suli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39, 2133–2163 (2002)

    Article  MathSciNet  Google Scholar 

  21. Johnson, C., Pitkaranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)

    Article  MathSciNet  Google Scholar 

  22. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)

    Article  Google Scholar 

  23. Le Veque, R.: Finite volume methods for hyperbolic problems. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

  24. Lesaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. Math. Asp. Finite Elem. Partial Differ. Equ. 33, 89–123 (1974)

    Article  MathSciNet  Google Scholar 

  25. Li, R., Chen, Z., Wu, W.: The Generalized Difference Methods for Partial Differential Equations (Numerical Analysis of Finite Volume Methods). Marcel Dikker, New York (2000)

  26. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)

    Article  MathSciNet  Google Scholar 

  27. Liu, Y., Vinokur, M., Wang, Z.: Spectral (finite) volume method for conservation laws on unstructured grids. V. Extension to three-dimensional systems. J. Comput. Phys. 212, 454–472 (2006)

  28. Peterson, T.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28, 133–140 (1991)

    Article  MathSciNet  Google Scholar 

  29. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM (1973)

  30. Richter, G.R.: An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50, 75–88 (1988)

    Article  MathSciNet  Google Scholar 

  31. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  Google Scholar 

  32. Sonar,T.: Classical finite volume methods. In: Handbook of Numerical Analysis, vol. XVII, pp. 55–76 (2016)

  33. Sun, Y., Wang, Z.: Evaluation of discontinuous Galerkin and spectral volumemethods for scalar and system conservation laws unconstructured grids. Int. J. Numer. Meth. Fluids 45, 819–838 (2004)

    Article  Google Scholar 

  34. Sun, Y., Wang, Z., Liu, Y.: High-order multidomain spectral difference method for the Navier–Stokes equations on unstructured hexahedral grids, Commun. Comput. Phys. 2, 310–333 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Sun, Y., Wang, Z., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids. VI. Extension to viscous flow. J. Comput. Phys. 215, 41–58 (2006)

  36. Van den Abeele, K., Lacor, C.: An accuracy and stability study of the 2D spectral volume method. J. Comput. Phys. 226, 1007–1026 (2007)

    Article  MathSciNet  Google Scholar 

  37. 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, 616–636 (2007)

    Article  MathSciNet  Google Scholar 

  38. Van den Abeele, K., Ghorbaniasl, G., Parsani, M., Lacor, C.: A stability analysis for the spectral volume method on tetrahedral grids. J. Comput. Phys. 228, 257–265 (2009)

    Article  Google Scholar 

  39. Van den Abeele, K., Lacor, C., Wang, Z.: On the connection between the spectral volume method and the spectral difference method. IV. Extension to two-dimensional systems. J. Comput. Phys. 227, 877–885 (2007)

  40. Wang, Z.: Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation. J. Comput. Phys. 178, 210–251 (2002)

  41. Wang, Z., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids, II. Extension to two-dimensional scalar equation. J. Comput. Phys. 179, 665–697 (2002)

  42. Wang, Z., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids. III. One dimensional systems and partition optimization. J. Sci. Comput. 20, 137–157 (2004)

  43. Wang, Z., Zhang, L., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids. IV. Extension to two-dimensional systems. J. Comput. Phys. 194, 716–741 (2004)

  44. Xu, J., Zou, Q.: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations. Numerische Mathematik 111, 469–492 (2009)

    Article  MathSciNet  Google Scholar 

  45. Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110–3133 (2012)

    Article  MathSciNet  Google Scholar 

  46. Zhang, M., Shu, C.-W.: An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids 34, 581–592 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Waixiang Cao

  2. School of Computer Science and Engineering, and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Qingsong Zou

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  1. Waixiang Cao
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Correspondence to Qingsong Zou.

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Waixiang Cao was supported in part by NSFC Grant No. 11871106.

Qingsong Zou was supported in part by NSFC Grant 12071496, Guangdong Provincial NSF Grant 2017B030311001, and Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(2020B1212060032).

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Cao, W., Zou, Q. Analysis of Spectral Volume Methods for 1D Linear Scalar Hyperbolic Equations. J Sci Comput 90, 61 (2022). https://doi.org/10.1007/s10915-021-01715-5

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  • DOI: https://doi.org/10.1007/s10915-021-01715-5

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