Spectral viscosity approximations to multidimensional scalar conservation laws

Authors:
Gui Qiang Chen, Qiang Du and Eitan Tadmor

Journal:
Math. Comp. **61** (1993), 629-643

MSC:
Primary 35L65; Secondary 65M12

MathSciNet review:
1185240

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Abstract: We study the spectral viscosity (SV) method in the context of multidimensional scalar conservation laws with periodic boundary conditions. We show that the spectral viscosity, which is sufficiently small to retain the formal spectral accuracy of the underlying Fourier approximation, is large enough to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Moreover, we prove that because of the presence of the spectral viscosity, the truncation error in this case becomes spectrally small, *independent* of whether the underlying solution is smooth or not. Consequently, the SV approximation remains uniformly bounded and converges to a measure-valued solution satisfying the entropy condition, that is, the unique entropy solution. We also show that the SV solution has a bounded total variation, provided that the total variation of the initial data is bounded, thus confirming its strong convergence to the entropy solution. We obtain an convergence rate of the usual optimal order one-half.

**[1]**C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,*Spectral methods*, Scientific Computation, Springer, Berlin, 2007. Evolution to complex geometries and applications to fluid dynamics. MR**2340254****[2]**G.-Q. Chen,*The compensated compactness method and the system of isentropic gas dynamics*, Preprint MSRI-00527-91, Mathematical Science Research Institute, Berkeley, October 1990.**[3]**Gui Qiang Chen,*The method of quasidecoupling for discontinuous solutions to conservation laws*, Arch. Rational Mech. Anal.**121**(1992), no. 2, 131–185. MR**1188491**, 10.1007/BF00375416**[4]**Michael G. Crandall and Andrew Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, 10.1090/S0025-5718-1980-0551288-3**[5]**Ronald J. DiPerna,*Measure-valued solutions to conservation laws*, Arch. Rational Mech. Anal.**88**(1985), no. 3, 223–270. MR**775191**, 10.1007/BF00752112**[6]**R. J. DiPerna,*Convergence of approximate solutions to conservation laws*, Arch. Rational Mech. Anal.**82**(1983), no. 1, 27–70. MR**684413**, 10.1007/BF00251724**[7]**David Gottlieb and Steven A. Orszag,*Numerical analysis of spectral methods: theory and applications*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. MR**0520152****[8]**David Gottlieb and Eitan Tadmor,*Recovering pointwise values of discontinuous data within spectral accuracy*, Progress and supercomputing in computational fluid dynamics (Jerusalem, 1984), Progr. Sci. Comput., vol. 6, Birkhäuser Boston, Boston, MA, 1985, pp. 357–375. MR**935160****[9]**Heinz-Otto Kreiss,*Fourier expansions of the solutions of the Navier-Stokes equations and their exponential decay rate*, Analyse mathématique et applications, Gauthier-Villars, Montrouge, 1988, pp. 245–262. MR**956963****[10]**Heinz-Otto Kreiss and Joseph Oliger,*Comparison of accurate methods for the integration of hyperbolic equations*, Tellus**24**(1972), 199–215 (English, with Russian summary). MR**0319382****[11]**N. N. Kuznetsov,*On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions*, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 183–197. MR**0657786****[12]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[13]**Yvon Maday and Eitan Tadmor,*Analysis of the spectral vanishing viscosity method for periodic conservation laws*, SIAM J. Numer. Anal.**26**(1989), no. 4, 854–870. MR**1005513**, 10.1137/0726047**[14]**Richard Sanders,*On convergence of monotone finite difference schemes with variable spatial differencing*, Math. Comp.**40**(1983), no. 161, 91–106. MR**679435**, 10.1090/S0025-5718-1983-0679435-6**[15]**S. Schochet,*The rate of convergence of spectral-viscosity methods for periodic scalar conservation laws*, SIAM J. Numer. Anal.**27**(1990), no. 5, 1142–1159. MR**1061123**, 10.1137/0727066**[16]**Denis Serre,*La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations à une dimension d’espace*, J. Math. Pures Appl. (9)**65**(1986), no. 4, 423–468 (French). MR**881690****[17]**Joel Smoller,*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146****[18]**Anders Szepessy,*Measure-valued solutions of scalar conservation laws with boundary conditions*, Arch. Rational Mech. Anal.**107**(1989), no. 2, 181–193. MR**996910**, 10.1007/BF00286499**[19]**Eitan Tadmor,*Convergence of spectral methods for nonlinear conservation laws*, SIAM J. Numer. Anal.**26**(1989), no. 1, 30–44. MR**977947**, 10.1137/0726003**[20]**Eitan Tadmor,*Shock capturing by the spectral viscosity method*, Comput. Methods Appl. Mech. Engrg.**80**(1990), no. 1-3, 197–208. Spectral and high order methods for partial differential equations (Como, 1989). MR**1067951**, 10.1016/0045-7825(90)90023-F**[21]**-,*Semi-discrete approximations to nonlinear systems of conservation laws*;*consistency and*-*stability imply convergence*, ICASE Report No. 88-41, 1988.**[22]**Eitan Tadmor,*Total variation and error estimates for spectral viscosity approximations*, Math. Comp.**60**(1993), no. 201, 245–256. MR**1153170**, 10.1090/S0025-5718-1993-1153170-9

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1993-1185240-3

Keywords:
Multidimensional conservation laws,
spectral viscosity method,
spectral accuracy,
measure-valued solution,
total variation,
convergence,
error estimate

Article copyright:
© Copyright 1993
American Mathematical Society