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15-05-2019 | Original Paper | Issue 3/2020

A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems

Journal:: Numerical Algorithms > Issue 3/2020

Author:: Naveen Kumar Garg

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Abstract

In this article, a class of upwind schemes is proposed for systems, each of which yields an incomplete set of linearly independent eigenvectors. The theory of Jordan canonical forms is used to complete such sets through the addition of generalized eigenvectors. A modified Burgers’ system and its extensions generate δ,δ^′, δ^″,⋯,δⁿ waves as solutions. The performance of flux difference splitting-based numerical schemes is examined by considering various numerical examples. Since the flux Jacobian matrix of pressureless gas dynamics system also produces an incomplete set of linearly independent eigenvectors, a similar framework is adopted to construct a numerical algorithm for a pressureless gas dynamics system.

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Berthon, C., Breuß, M., Titeux, M.O.: A relaxation scheme for the approximation of the pressureless Euler equations. Numer. Methods Partial Differ. Equ. 22, 484–505 (2006). https://doi.org/10.1002/num.20108 MathSciNetCrossRef

Bouchut, F., James, F.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. Partial Differ. Equ. 24(11-12), 2173–2189 (1999). https://doi.org/10.1080/03605309908821498 MathSciNetCrossRef

Bouchut, F., Jin, S., Li, X.: Numerical approximations of pressureless and isothermal gas dynamics. SIAM J. Numer. Anal. 41(1), 135–158 (2003). https://doi.org/10.1137/S0036142901398040 MathSciNetCrossRef

Cockburn, B.: Discontinuous Galerkin methods for convection-dominated problems, High-order methods for computational physics. Lect. Notes Comput. Sci. Eng. 9, 69–224 (1999). https://doi.org/10.1007/978-3-662-03882-6_2 CrossRef

Cockburn, B., Karniadakis, G.E., Shu, C.W.: The development of discontinuous Galerkin methods, Discontinuous Galerkin methods (Newport, RI, 1999). Lect. Notes Comput. Sci. Eng. 11, 3–50 (2000). https://doi.org/10.1007/978-3-642-59721-3_1 CrossRef

Engquist, B., Runborg, O.: Multi-phase computations in geometrical optics. J. Comput. Appl. Math. 74, 175–192 (1996). https://doi.org/10.1016/0377-0427(96)00023-4 MathSciNetCrossRef

Friedrichs, K.O.: Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7, 345–392 (1954). https://doi.org/10.1002/cpa.3160070206 MathSciNetCrossRef

Garg, N.K., Junk, M., Rao, S., Sekhar, M.: An upwind method for genuine weakly hyperbolic systems, arXiv: http://arXiv.org/abs/1703.08751 (2017)

Garg, N.K., Rao, S., Sekhar, M.: An Approximate Riemann Solver for Convection-Pressure Split Euler Equations Using Jordan Canonical Forms, arXiv: http://arXiv.org/abs/1607.00947v1 (2016)

10.

Garg, N.K., Rao, S., Sekhar, M.: Use of Jordan forms for convection-pressure split Euler solvers. arXiv: http://arXiv.org/abs/1607.00947v5 (2017)

11.

Harten, A.: On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21, 1–23 (1984). https://doi.org/10.1137/0721001 MathSciNetCrossRef

12.

Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions. II. Entropy production at shocks. J. Comput. Phys. 228, 5410–5436 (2009). https://doi.org/10.1016/j.jcp.2009.04.021 MathSciNetCrossRef

13.

Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996). https://doi.org/10.1006/jcph.1996.0130 MathSciNetCrossRef

14.

Jiang, G.S., Tadmor, E.: Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19, 1892–1917 (1998). https://doi.org/10.1137/S106482759631041X MathSciNetCrossRef

15.

Joseph, K.T.: A Riemann problem whose viscosity solutions contain δ-measures. Asymptot. Anal. 7, 105–120 (1993) MathSciNetCrossRef

16.

Joseph, K.T.: Explicit generalized solutions to a system of conservation laws. Proc. Indian Acad. Sci. Math. Sci. 109(4), 401–409 (1999). https://doi.org/10.1007/BF02838000 MathSciNetCrossRef

17.

Joseph, K.T., Sahoo, M.R.: Vanishing viscosity approach to a system of conservation laws admitting δ ^″ waves. Commun. Pure Appl. Anal. 12 (5), 2091–2118 (2013). https://doi.org/10.3934/cpaa.2013.12.2091 MathSciNetCrossRef

18.

Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Anal. 7, 159–193 (1954). https://doi.org/10.1002/cpa.3160070112 MathSciNetCrossRef

19.

LeVeque, R.J.: The dynamics of pressureless dust clouds and delta waves. J. Hyperbolic Differ. Equ. 1 (2), 315–327 (2004). https://doi.org/10.1142/S0219891604000135 MathSciNetCrossRef

20.

Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994). https://doi.org/10.1006/jcph.1994.1187 MathSciNetCrossRef

21.

Liu, X.D., Tadmor, E.: Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math. 79, 397–425 (1998). https://doi.org/10.1007/s002110050345 MathSciNetCrossRef

22.

Panov, E.Y., Shelkovich, V.M.: δ ^′-shock waves as a new type of solutions to systems of conservation laws. J. Diff. Equat. 228, 49–86 (2006). https://doi.org/10.1016/j.jde.2006.04.004 MathSciNetCrossRef

23.

Parés, C., Castro, M.: On the well-balance property of Roe’s method for nonconservative hyperbolic systems, Applications to shallow-water systems. M2AN Math. Model. Numer. Anal. 38, 821–852 (2004). https://doi.org/10.1051/m2an:2004041 MathSciNetCrossRef

24.

Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981). https://doi.org/10.1016/0021-9991(81)90128-5 MathSciNetCrossRef

25.

Rusanov, V.V.E.: The calculation of the interaction of non-stationary shock waves with barriers, ž. vyčisl. Mat. i Mat Fiz. 1, 267–279 (1961) MathSciNet

26.

Shelkovich, V.M.: The Riemann problem admitting δ-, δ ^′-shocks, and vacuum states (the vanishing viscosity approach). J. Diff. Equat. 231(2), 459–500 (2006). https://doi.org/10.1016/j.jde.2006.08.003 MathSciNetCrossRef

27.

Sheng, W., Zhang, T.: The Riemann problem for the transportation equations in gas dynamics. Mem. Amer. Math. Soc. 137, viii+ 77 (1999). https://doi.org/10.1090/memo/0654 MathSciNetMATH

28.

Smith, T.A., Petty, D.J., Pantano, C.: A Roe-like numerical method for weakly hyperbolic systems of equations in conservation and non-conservation form. J. Comput. Phys. 316, 117–138 (2016). https://doi.org/10.1016/j.jcp.2016.04.006 MathSciNetCrossRef

29.

Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws I. Math. Comp. 49, 91–103 (1987). https://doi.org/10.2307/2008251 MathSciNetCrossRef

30.

Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156 MathSciNetCrossRef

31.

Toro, E.F.: Shock-capturing methods for free-surface shallow flows. Wiley, New York (2001) MATH

32.

Toro, E.F., Vázquez-Cendón, M.E.: Flux splitting schemes for the Euler equations. Comput. Fluids 70, 1–12 (2012). https://doi.org/10.1016/j.compfluid.2012.08.023 MathSciNetCrossRef

33.

Yang, Y., Wei, D., Shu, C.W.: Discontinuous Galerkin method for Krause’s consensus models and pressureless Euler equations. J. Comput. Phys. 252, 109–127 (2013). https://doi.org/10.1016/j.jcp.2013.06.015 MathSciNetCrossRef

34.

Zel’Dovich, Y.B.: Gravitational instability: an approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89 (1970)

Title: A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems
Author:: Naveen Kumar Garg
Publication date: 15-05-2019
DOI: https://doi.org/10.1007/s11075-019-00717-7
Publisher: Springer US
Journal: Numerical Algorithms
Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265

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