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
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
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
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
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
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
Friedrichs, K.O.: Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7, 345–392 (1954). https://doi.org/10.1002/cpa.3160070206
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)
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)
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
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
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
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
Joseph, K.T.: A Riemann problem whose viscosity solutions contain δ-measures. Asymptot. Anal. 7, 105–120 (1993)
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
Toro, E.F.: Shock-capturing methods for free-surface shallow flows. Wiley, New York (2001)
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
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
Zel’Dovich, Y.B.: Gravitational instability: an approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89 (1970)
Acknowledgements
The author would like to thank Prof. G. D. Veerappa Gowda, Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore, and Prof. S. V. Raghurama Rao, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, for the fruitful discussions.
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Garg, N.K. A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems. Numer Algor 83, 1091–1121 (2020). https://doi.org/10.1007/s11075-019-00717-7
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DOI: https://doi.org/10.1007/s11075-019-00717-7