Why nonconservative schemes converge to wrong solutions: error analysis
Authors:
Thomas Y. Hou and Philippe G. LeFloch
Journal:
Math. Comp. 62 (1994), 497530
MSC:
Primary 65M12; Secondary 35L65, 65G05
MathSciNet review:
1201068
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Abstract: This paper attempts to give a qualitative and quantitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws. We show that these schemes converge strongly in norm to the solution of an inhomogeneous conservation law containing a Borel measure source term. Moreover, we analyze the properties of this Borel measure, and derive a sharp estimate for the error between the limit function given by the scheme and the correct solution. In general, the measure source term is of the order of the entropy dissipation measure associated with the scheme. In certain cases, the error can be small for short times, which makes it difficult to detect numerically. But generically, such an error will grow in time, and this would lead to a large error for largetime calculations. Finally, we show that a local correction of any highorder accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution.
 [1]
J. F. Colombeau and A. Y. Leroux, Numerical methods for hyperbolic systems in nonconservative form using products of distributions, Advances in Computer Methods for P.D.E., vol. 6 (R. Vichnevetsky and R. S. Stepleman, eds.), Inst. Math. Appl. Conf. Series, Oxford Univ. Press, New York, 1987, pp. 2837.
 [2]
Frédéric
Coquel and Philippe
LeFloch, Convergence of finite difference
schemes for conservation laws in several space dimensions: the corrected
antidiffusive flux approach, Math. Comp.
57 (1991), no. 195, 169–210. MR 1079010
(91m:65229), http://dx.doi.org/10.1090/S00255718199110790102
 [3]
, On the finite volume method for multidimensional conservation laws, preprint, December 1991, Courant Institute, New York University (unpublished).
 [4]
Michael
G. Crandall and Andrew
Majda, Monotone difference approximations for
scalar conservation laws, Math. Comp.
34 (1980), no. 149, 1–21. MR 551288
(81b:65079), http://dx.doi.org/10.1090/S00255718198005512883
 [5]
C.
M. Dafermos, Characteristics in hyperbolic conservation laws. A
study of the structure and the asymptotic behaviour of solutions,
Nonlinear analysis and mechanics: HeriotWatt Symposium (Edinburgh, 1976),
Vol. I, Pitman, London, 1977, pp. 1–58. Res. Notes in Math.,
No. 17. MR
0481581 (58 #1693)
 [6]
Gianni
Dal Maso, Philippe
G. Lefloch, and François
Murat, Definition and weak stability of nonconservative
products, J. Math. Pures Appl. (9) 74 (1995),
no. 6, 483–548. MR 1365258
(97b:46052)
 [7]
Ronald
J. DiPerna, Finite difference schemes for conservation laws,
Comm. Pure Appl. Math. 35 (1982), no. 3,
379–450. MR
649350 (84h:35100), http://dx.doi.org/10.1002/cpa.3160350305
 [8]
Ronald
J. DiPerna and Andrew
Majda, The validity of nonlinear geometric optics for weak
solutions of conservation laws, Comm. Math. Phys. 98
(1985), no. 3, 313–347. MR 788777
(87e:35057)
 [9]
B. Engquist and B. Sjogreen, Numerical approximation of hyperbolic conservation laws with stiff terms, UCLA Computational and Applied Mathematics Report 8907, 1989.
 [10]
James
Glimm, Solutions in the large for nonlinear hyperbolic systems of
equations, Comm. Pure Appl. Math. 18 (1965),
697–715. MR 0194770
(33 #2976)
 [11]
James
Glimm and Peter
D. Lax, Decay of solutions of systems of nonlinear hyperbolic
conservation laws, Memoirs of the American Mathematical Society, No.
101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
(42 #676)
 [12]
Jonathan
Goodman and Peter
D. Lax, On dispersive difference schemes. I, Comm. Pure Appl.
Math. 41 (1988), no. 5, 591–613. MR 948073
(89f:65094), http://dx.doi.org/10.1002/cpa.3160410506
 [13]
Eduard
Harabetian and Robert
Pego, Nonconservative hybrid shock capturing schemes, J.
Comput. Phys. 105 (1993), no. 1, 1–13. MR 1210856
(93m:65118), http://dx.doi.org/10.1006/jcph.1993.1048
 [14]
Ami
Harten, High resolution schemes for hyperbolic conservation
laws, J. Comput. Phys. 49 (1983), no. 3,
357–393. MR
701178 (84g:65115), http://dx.doi.org/10.1016/00219991(83)901365
 [15]
Ami
Harten and Stanley
Osher, Uniformly highorder accurate nonoscillatory schemes.
I, SIAM J. Numer. Anal. 24 (1987), no. 2,
279–309. MR
881365 (90a:65198), http://dx.doi.org/10.1137/0724022
 [16]
A.
Harten and G.
Zwas, Selfadjusting hybrid schemes for shock computations, J.
Computational Phys. 9 (1972), 568–583. MR 0309339
(46 #8449)
 [17]
Thomas
Y. Hou and Peter
D. Lax, Dispersive approximations in fluid dynamics, Comm.
Pure Appl. Math. 44 (1991), no. 1, 1–40. MR 1077912
(91m:76088), http://dx.doi.org/10.1002/cpa.3160440102
 [18]
S.
Karni, Viscous shock profiles and primitive formulations, SIAM
J. Numer. Anal. 29 (1992), no. 6, 1592–1609. MR 1191138
(93j:65163), http://dx.doi.org/10.1137/0729092
 [19]
S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSRSb. 10 (1970), 217243.
 [20]
N. N. Kuznetsov, Accuracy of some approximate method for computing the weak solutions of a first order quasilinear equation, USSR Comput. Math. and Math. Phys. 16 (1976), 105119.
 [21]
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
(50 #2709)
 [22]
Peter
Lax and Burton
Wendroff, Systems of conservation laws, Comm. Pure Appl. Math.
13 (1960), 217–237. MR 0120774
(22 #11523)
 [23]
Ph. Le Floch and J.G Liu, Entropy and monotonicity (EMO) consistent schemes for conservation laws, preprint, 1993.
 [24]
Philippe
LeFloch and TaiPing
Liu, Existence theory for nonlinear hyperbolic systems in
nonconservative form, Forum Math. 5 (1993),
no. 3, 261–280. MR 1216035
(94e:35086), http://dx.doi.org/10.1515/form.1993.5.261
 [25]
A.
Y. LeRoux and P.
Quesseveur, Convergence of an antidiffusion LagrangeEuler scheme
for quasilinear equations, SIAM J. Numer. Anal. 21
(1984), no. 5, 985–994. MR 760627
(85m:65092), http://dx.doi.org/10.1137/0721061
 [26]
Gino
Moretti, The 𝜆scheme, Comput. & Fluids
7 (1979), no. 3, 191–205. MR 549494
(80j:76045), http://dx.doi.org/10.1016/00457930(79)900367
 [27]
Eitan
Tadmor, Local error estimates for discontinuous solutions of
nonlinear hyperbolic equations, SIAM J. Numer. Anal.
28 (1991), no. 4, 891–906. MR 1111445
(92d:35190), http://dx.doi.org/10.1137/0728048
 [28]
J. A. Trangenstein, A secondorder algorithm for the dynamic response of soils, Impact Comput. Sci. Engrg. 2 (1990), 139.
 [29]
John
A. Trangenstein and Phillip
Colella, A higherorder Godunov method for modeling finite
deformation in elasticplastic solids, Comm. Pure Appl. Math.
44 (1991), no. 1, 41–100. MR 1077913
(92f:73029), http://dx.doi.org/10.1002/cpa.3160440103
 [30]
A.
I. Vol′pert, Spaces 𝐵𝑉 and quasilinear
equations, Mat. Sb. (N.S.) 73 (115) (1967),
255–302 (Russian). MR 0216338
(35 #7172)
 [31]
G. Zwas and J. Roseman, The effect of nonlinear transformations on the computation of weak solutions, J. Comput. Phys. 12 (1973), 179186.
 [1]
 J. F. Colombeau and A. Y. Leroux, Numerical methods for hyperbolic systems in nonconservative form using products of distributions, Advances in Computer Methods for P.D.E., vol. 6 (R. Vichnevetsky and R. S. Stepleman, eds.), Inst. Math. Appl. Conf. Series, Oxford Univ. Press, New York, 1987, pp. 2837.
 [2]
 F. Coquel and Ph. Le Floch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), 169210. MR 1079010 (91m:65229)
 [3]
 , On the finite volume method for multidimensional conservation laws, preprint, December 1991, Courant Institute, New York University (unpublished).
 [4]
 M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 121. MR 551288 (81b:65079)
 [5]
 C. M. Dafermos, Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behavior of solutions (R. J. Knops, ed.), HeriotWatt University, Nonlinear Analysis and Mechanics: HeriotWatt Symposium Volume I, 1981, pp. 158. MR 0481581 (58:1693)
 [6]
 G. Dal Maso, Ph. Le Floch, and F. Murat, Definition and weak stability of nonconservative products, Preprint CMAP, Ecole Polytechnique, Palaiseau (France). MR 1365258 (97b:46052)
 [7]
 R. DiPerna, Finite difference scheme for conservation laws, Comm. Pure Appl. Math. 25 (1982), 379450. MR 649350 (84h:35100)
 [8]
 R. J. DiPerna and A. Majda, The validity of nonlinear geometric optics for weak solutions of conservation laws, Comm. Math. Phys. 98 (1985), 313347. MR 788777 (87e:35057)
 [9]
 B. Engquist and B. Sjogreen, Numerical approximation of hyperbolic conservation laws with stiff terms, UCLA Computational and Applied Mathematics Report 8907, 1989.
 [10]
 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of conservation laws, Comm. Pure Appl. Math. 18 (1965), 695715. MR 0194770 (33:2976)
 [11]
 J. Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. No. 101, Amer. Math. Soc., Providence, RI, 1970. MR 0265767 (42:676)
 [12]
 J. Goodman and P. D. Lax, On dispersive difference schemes, Comm. Pure Appl. Math. 41 (1988), 591613. MR 948073 (89f:65094)
 [13]
 E. Harabetian and R. Pego, Efficient hybrid shock capturing scheme, IMA Preprint Series No. 743, Minneapolis, MN, 1990. MR 1210856 (93m:65118)
 [14]
 A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), 357393. MR 701178 (84g:65115)
 [15]
 A. Harten and S. Osher, Uniformly high order accurate nonoscillatory schemes I, SIAM J. Numer. Anal. 24 (1987), 279309. MR 881365 (90a:65198)
 [16]
 A. Harten and G. Zwas, Selfadjusting fluid schemes for shock computations, J. Comput. Phys. 9 (1972), 568583. MR 0309339 (46:8449)
 [17]
 T. Y. Hou and P. D. Lax, Dispersive approximations in fluid dynamics, Comm. Pure Appl. Math. 44 (1991), 140. MR 1077912 (91m:76088)
 [18]
 S. Kami, Viscous shock profiles and primitive formulations, SIAM J. Numer. Anal. 29 (1992), 15921609. MR 1191138 (93j:65163)
 [19]
 S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSRSb. 10 (1970), 217243.
 [20]
 N. N. Kuznetsov, Accuracy of some approximate method for computing the weak solutions of a first order quasilinear equation, USSR Comput. Math. and Math. Phys. 16 (1976), 105119.
 [21]
 P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS Monographs, vol. 11, SIAM, Philadelphia, PA, 1973. MR 0350216 (50:2709)
 [22]
 P. D. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217237. MR 0120774 (22:11523)
 [23]
 Ph. Le Floch and J.G Liu, Entropy and monotonicity (EMO) consistent schemes for conservation laws, preprint, 1993.
 [24]
 Ph. Le Floch and T.P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), 261280. MR 1216035 (94e:35086)
 [25]
 A. Y. Leroux and P. Quesseveur, Convergence of an antidiffusive LagrangeEuler scheme for quasilinear equations, SIAM J. Numer. Anal. 21 (1984), 985994. MR 760627 (85m:65092)
 [26]
 G. Moretti, The scheme, Comput. & Fluids 7 (1979), 191205. MR 549494 (80j:76045)
 [27]
 E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891906. MR 1111445 (92d:35190)
 [28]
 J. A. Trangenstein, A secondorder algorithm for the dynamic response of soils, Impact Comput. Sci. Engrg. 2 (1990), 139.
 [29]
 J. A. Trangenstein and P. Colella, A higherorder Godunov method for modeling finite deformations in elasticplastic solids, Comm. Pure Appl. Math. 44 (1991), 41100. MR 1077913 (92f:73029)
 [30]
 A. I. Volpert, The space BV and quasilinear equations, Math. USSR Sb. 2 (1967), 225267. MR 0216338 (35:7172)
 [31]
 G. Zwas and J. Roseman, The effect of nonlinear transformations on the computation of weak solutions, J. Comput. Phys. 12 (1973), 179186.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199412010680
PII:
S 00255718(1994)12010680
Keywords:
Hyperbolic conservation law,
entropy discontinuous solution,
nonconservative scheme,
numerical error
Article copyright:
© Copyright 1994
American Mathematical Society
