Convergence of upwind schemes for a stationary shock

Author:
Jens Lorenz

Journal:
Math. Comp. **46** (1986), 45-57

MSC:
Primary 65M10; Secondary 76-08, 76L05

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815830-2

MathSciNet review:
815830

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Abstract: A nonlinear first-order 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 EO-scheme, 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**, https://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**, https://doi.org/10.1090/S0025-5718-1980-0551288-3**[3]**Pedro Embid, Jonathan Goodman, and Andrew Majda,*Multiple steady states for 1-D transonic flow*, SIAM J. Sci. Statist. Comput.**5**(1984), no. 1, 21–41. MR**731879**, https://doi.org/10.1137/0905002**[4]**Björn Engquist and Stanley Osher,*One-sided difference approximations for nonlinear conservation laws*, Math. Comp.**36**(1981), no. 154, 321–351. MR**606500**, https://doi.org/10.1090/S0025-5718-1981-0606500-X**[5]**F. A. Howes,*Boundary-interior layer interactions in nonlinear singular perturbation theory*, Mem. Amer. Math. Soc.**15**(1978), no. 203, iv+108. MR**0499407**, https://doi.org/10.1090/memo/0203**[6]**Lloyd K. Jackson,*Subfunctions and second-order ordinary differential inequalities*, Advances in Math.**2**(1968), 307–363. MR**0229896**, https://doi.org/10.1016/0001-8708(68)90022-4**[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****[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, Berlin-New York, 1982, pp. 150–169. MR**679352****[9]**J. Lorenz,*Numerical solution of a singular perturbation problem with turning points*, Equadiff 82 (Würzburg, 1982) Lecture Notes in Math., vol. 1017, Springer, Berlin-New York, 1983, pp. 432–439. MR**726601****[10]**Jens Lorenz,*Analysis of difference schemes for a stationary shock problem*, SIAM J. Numer. Anal.**21**(1984), no. 6, 1038–1053. MR**765505**, https://doi.org/10.1137/0721064**[11]**M. Nagumo, "Über die Differentialgleichung ,"*Proc. Math. Soc. Japan*, v. 19, 1937, pp. 861-866.**[12]**O. A. Oleĭnik,*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737****[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) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 179–204. MR**605507****[14]**Stanley Osher,*Nonlinear singular perturbation problems and one-sided difference schemes*, SIAM J. Numer. Anal.**18**(1981), no. 1, 129–144. MR**603435**, https://doi.org/10.1137/0718010**[15]**G. R. Shubin, A. B. Stephens & H. M. Glaz, "Steady shock tracking and Newton's method applied to one-dimensional duct flow,"*J. Comput. Phys.*, v. 39, 1981, pp. 364-374.**[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**, https://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.

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0815830-2

Article copyright:
© Copyright 1986
American Mathematical Society