Convergence of upwind schemes for a stationary shock
Author:
Jens Lorenz
Journal:
Math. Comp. 46 (1986), 4557
MSC:
Primary 65M10; Secondary 7608, 76L05
MathSciNet review:
815830
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A nonlinear firstorder boundary value problem with discontinuous solutions is considered. It arises in the study of gasflow through a duct and allows, in general, for multiple solutions. New convergence results for three difference schemes are presented and the sharpness of numerical layers is established. For the EOscheme, stability of a physically correct solution with respect to time evolution is shown.
 [1]
C.
Bardos, A.
Y. le Roux, and J.C.
Nédélec, First order quasilinear equations with
boundary conditions, Comm. Partial Differential Equations
4 (1979), no. 9, 1017–1034. MR 542510
(81b:35052), http://dx.doi.org/10.1080/03605307908820117
 [2]
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
 [3]
Pedro
Embid, Jonathan
Goodman, and Andrew
Majda, Multiple steady states for 1D transonic flow, SIAM J.
Sci. Statist. Comput. 5 (1984), no. 1, 21–41.
MR 731879
(86a:76029), http://dx.doi.org/10.1137/0905002
 [4]
Björn
Engquist and Stanley
Osher, Onesided difference approximations
for nonlinear conservation laws, Math.
Comp. 36 (1981), no. 154, 321–351. MR 606500
(82c:65056), http://dx.doi.org/10.1090/S0025571819810606500X
 [5]
F.
A. Howes, Boundaryinterior layer interactions in nonlinear
singular perturbation theory, Mem. Amer. Math. Soc.
15 (1978), no. 203, iv+108. MR 0499407
(58 #17288)
 [6]
Lloyd
K. Jackson, Subfunctions and secondorder ordinary differential
inequalities, Advances in Math. 2 (1968),
307–363 (1968). MR 0229896
(37 #5462)
 [7]
H.
W. Liepmann and A.
Roshko, Elements of gasdynamics, Galcit aeronautical series,
John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London,
1957. MR
0092515 (19,1121a)
 [8]
Jens
Lorenz, Nonlinear boundary value problems with turning points and
properties of difference schemes, Theory and applications of singular
perturbations (Oberwolfach, 1981), Lecture Notes in Math., vol. 942,
Springer, BerlinNew York, 1982, pp. 150–169. MR 679352
(84a:34025)
 [9]
J.
Lorenz, Numerical solution of a singular perturbation problem with
turning points, Equadiff 82 (Würzburg, 1982) Lecture Notes in
Math., vol. 1017, Springer, BerlinNew York, 1983,
pp. 432–439. MR 726601
(84m:65090)
 [10]
Jens
Lorenz, Analysis of difference schemes for a stationary shock
problem, SIAM J. Numer. Anal. 21 (1984), no. 6,
1038–1053. MR 765505
(86b:34107), http://dx.doi.org/10.1137/0721064
 [11]
M. Nagumo, "Über die Differentialgleichung ," Proc. Math. Soc. Japan, v. 19, 1937, pp. 861866.
 [12]
O.
A. Oleĭnik, Discontinuous solutions of nonlinear
differential equations, Amer. Math. Soc. Transl. (2)
26 (1963), 95–172. MR 0151737
(27 #1721)
 [13]
Stanley
Osher, Numerical solution of singular perturbation problems and
hyperbolic systems of conservation laws, Analytical and numerical
approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen,
Nijmegen, 1980) NorthHolland Math. Stud., vol. 47, NorthHolland,
AmsterdamNew York, 1981, pp. 179–204. MR 605507
(83g:65098)
 [14]
Stanley
Osher, Nonlinear singular perturbation problems and onesided
difference schemes, SIAM J. Numer. Anal. 18 (1981),
no. 1, 129–144. MR 603435
(83c:65188), http://dx.doi.org/10.1137/0718010
 [15]
G. R. Shubin, A. B. Stephens & H. M. Glaz, "Steady shock tracking and Newton's method applied to onedimensional duct flow," J. Comput. Phys., v. 39, 1981, pp. 364374.
 [16]
A.
B. Stephens and G.
R. Shubin, Existence and uniqueness for an exponentially derived
switching scheme, SIAM J. Numer. Anal. 20 (1983),
no. 5, 885–889. MR 714686
(85c:65109), http://dx.doi.org/10.1137/0720061
 [17]
A. B. Stephens & G. R. Shubin, Exponentially Derived Switching Schemes for Inviscid Flow, Applied Math. Branch, Naval Surface Weapons Center, Silver Spring, Maryland, Report, 1981.
 [1]
 C. Bardos, A. Y. LeRoux and J. C. Nedelec, "Firstorder quasilinear equations with boundary conditions," Comm. Partial Differential Equations, v. 4, 1979, pp. 10171034. MR 542510 (81b:35052)
 [2]
 M. G. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [3]
 P. Embid, J. Goodman & A. Majda, "Multiple steady states for 1D transonic flow," SIAM J. Sci. Statist. Comput., v. 5, 1984, pp. 2141. MR 731879 (86a:76029)
 [4]
 B. Engquist & S. Osher, "Onesided difference approximations for nonlinear conservation laws," Math. Comp., v. 36, 1981, pp. 321351. MR 606500 (82c:65056)
 [5]
 F. A. Howes, "Boundaryinterior layer interactions in nonlinear singular perturbation theory," Mem. Amer. Math. Soc., No. 203, 1978. MR 0499407 (58:17288)
 [6]
 L. K. Jackson, "Subfunctions and secondorder ordinary differential inequalities," Adv. in Math., v. 2, 1968, pp. 307363. MR 0229896 (37:5462)
 [7]
 H. W. Liepmann & A. Roshko, Elements of Gas Dynamics, Wiley, New York, 1957. MR 0092515 (19:1121a)
 [8]
 J. Lorenz, Nonlinear Boundary Value Problems with Turning Points and Properties of Difference Schemes (W. Eckhaus and E. M. de Jager, eds.), Springer Lecture Notes in Math., vol. 942, SpringerVerlag, Berlin and New York, 1982. MR 679352 (84a:34025)
 [9]
 J. Lorenz, "Numerical solution of a singular perturbation problem with turning points," in Equadiff 82 (H. W. Knobloch and K. Schmitt, eds.), Lecture Notes in Math., vol. 1017, SpringerVerlag, Berlin and New York, 1983. MR 726601 (84m:65090)
 [10]
 J. Lorenz, "Analysis of difference schemes for a stationary shock problem," SIAM J. Numer. Anal., v. 21, 1984, pp. 10381052. MR 765505 (86b:34107)
 [11]
 M. Nagumo, "Über die Differentialgleichung ," Proc. Math. Soc. Japan, v. 19, 1937, pp. 861866.
 [12]
 O. A. Oleinik, "Discontinuous solutions of nonlinear differential equations," Amer. Math. Soc. Transl. (2), v. 26, 1963, pp. 95172. MR 0151737 (27:1721)
 [13]
 S. Osher, "Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws," in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (O. Axelsson, L. S. Frank and A. von der Sluis, eds.), NorthHolland, Amsterdam, 1981. MR 605507 (83g:65098)
 [14]
 S. Osher, "Nonlinear singular perturbation problems and one sided difference schemes," SIAM J. Numer. Anal., v. 18, 1981, pp. 129144. MR 603435 (83c:65188)
 [15]
 G. R. Shubin, A. B. Stephens & H. M. Glaz, "Steady shock tracking and Newton's method applied to onedimensional duct flow," J. Comput. Phys., v. 39, 1981, pp. 364374.
 [16]
 A. B. Stephens & G. R. Shubin, "Existence and uniqueness for an exponentially derived switching scheme," SIAM J. Numer. Anal., v. 20, 1983, pp. 885889. MR 714686 (85c:65109)
 [17]
 A. B. Stephens & G. R. Shubin, Exponentially Derived Switching Schemes for Inviscid Flow, Applied Math. Branch, Naval Surface Weapons Center, Silver Spring, Maryland, Report, 1981.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M10,
7608,
76L05
Retrieve articles in all journals
with MSC:
65M10,
7608,
76L05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158302
PII:
S 00255718(1986)08158302
Article copyright:
© Copyright 1986
American Mathematical Society
