A finite element method for firstorder hyperbolic systems
Author:
Mitchell Luskin
Journal:
Math. Comp. 35 (1980), 10931112
MSC:
Primary 65N30; Secondary 65M15, 76N15
MathSciNet review:
583489
Fulltext PDF Free Access
Abstract 
References 
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Abstract: A new finite element method is proposed for the numerical solution of a class of initialboundary value problems for firstorder hyperbolic systems in one space dimension. An application of our procedure to a system modeling gas flow in a pipe is discussed. Asymptotic error estimates are derived in the norm in space.
 [1]
Jim
Douglas Jr., Todd
Dupont, and Lars
Wahlbin, Optimal 𝐿_{∞} error
estimates for Galerkin approximations to solutions of twopoint boundary
value problems, Math. Comp. 29 (1975), 475–483. MR 0371077
(51 #7298), http://dx.doi.org/10.1090/S00255718197503710770
 [2]
Todd
Dupont, Galerkin methods for modeling gas pipelines,
Constructive and computational methods for differential and integral
equations (Sympos., Indiana Univ., Bloomington, Ind., 1974) Springer,
Berlin, 1974, pp. 112–130. Lecture Notes in Math., Vol. 430. MR 0502035
(58 #19223)
 [3]
Todd
Dupont, Galerkin methods for first order hyperbolics: an
example, SIAM J. Numer. Anal. 10 (1973),
890–899. MR 0349046
(50 #1540)
 [4]
T. DUPONT & L. WAHLBIN, " optimality of weighted projections into piecewise polynomial spaces," Manuscript, Dept. of Math., Univ. of Chicago, 1974.
 [5]
G.
W. Hedstrom, The Galerkin method based on Hermite cubics, SIAM
J. Numer. Anal. 16 (1979), no. 3, 385–393. MR 530476
(80i:65116), http://dx.doi.org/10.1137/0716032
 [6]
Milton
Lees, A linear threelevel difference scheme
for quasilinear parabolic equations, Math.
Comp. 20 (1966),
516–522. MR 0207224
(34 #7040), http://dx.doi.org/10.1090/S00255718196602072245
 [7]
Mitchell
Luskin, On the existence of global smooth solutions for a model
equation for fluid flow in a pipe, J. Math. Anal. Appl.
84 (1981), no. 2, 614–630. MR 639688
(83g:76078), http://dx.doi.org/10.1016/0022247X(81)90192X
 [8]
M. LUSKIN, "A finite element method for first order hyperbolic systems in two space variables," Manuscript, Dept. of Math., Univ. of Michigan, 1978.
 [9]
G. PLATZMAN, "Normal modes of the world ocean. Part 1. Design of a finiteelement barotropic model," J. Phys. Oceanogr., v. 8, 1979, pp. 323343.
 [10]
H.
H. Rachford Jr., Twolevel discretetime Galerkin approximations
for second order nonlinear parabolic partial differential equations,
SIAM J. Numer. Anal. 10 (1973), 1010–1026. MR 0339519
(49 #4277)
 [11]
Gilbert
Strang and George
J. Fix, An analysis of the finite element method,
PrenticeHall, Inc., Englewood Cliffs, N. J., 1973. PrenticeHall Series in
Automatic Computation. MR 0443377
(56 #1747)
 [12]
V. STREETER, Fluid Mechanics, 5th ed., McGrawHill, New York, 1971.
 [13]
Lars
B. Wahlbin, A dissipative Galerkin method applied to some
quasilinear hyperbolic equations, Rev. Française Automat.
Informat. Recherche Opérationnelle Sér. Rouge
8 (1974), no. R2, 109–117 (English, with Loose
French summary). MR 0368447
(51 #4688)
 [1]
 J. DOUGLAS, JR., T. DUPONT & L. WAHLBIN, "Optimal error estimates for Galerkin approximations to solutions of twopoint boundary value problems," Math. Comp., v. 29, 1975, pp. 475483. MR 0371077 (51:7298)
 [2]
 T. DUPONT, "Galerkin methods for modeling gas pipelines," Constructive and Computational Methods for Differential and Integral Equations, Lecture Notes in Math., Vol. 430, SpringerVerlag, Berlin and New York, 1974. MR 0502035 (58:19223)
 [3]
 T. DUPONT, "Galerkin methods for first order hyperbolics: an example," SIAM J. Numer. Anal., v. 10, 1973, pp. 890899. MR 0349046 (50:1540)
 [4]
 T. DUPONT & L. WAHLBIN, " optimality of weighted projections into piecewise polynomial spaces," Manuscript, Dept. of Math., Univ. of Chicago, 1974.
 [5]
 G. HEDSTROM, "The Galerkin method based on Hermite cubics," SIAM J. Numer. Anal., v. 16, 1979, pp. 385393. MR 530476 (80i:65116)
 [6]
 M. LEES, "A linear three level difference scheme for quasilinear parabolic equations," Math. Comp., v. 20, 1966, pp. 516522. MR 0207224 (34:7040)
 [7]
 M. LUSKIN, "On the existence of global smooth solutions for a model equation for fluid flow in a pipe," Manuscript, Dept. of Math., Ecole Polytechnique Fédérale de Lausanne, 1980. MR 639688 (83g:76078)
 [8]
 M. LUSKIN, "A finite element method for first order hyperbolic systems in two space variables," Manuscript, Dept. of Math., Univ. of Michigan, 1978.
 [9]
 G. PLATZMAN, "Normal modes of the world ocean. Part 1. Design of a finiteelement barotropic model," J. Phys. Oceanogr., v. 8, 1979, pp. 323343.
 [10]
 H. RACHFORD, JR., "Twolevel discretetime Galerkin approximations for second order nonlinear parabolic partial differential equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 10101026. MR 0339519 (49:4277)
 [11]
 G. STRANG & G. FIX, An Analysis of the Finite Element Method, PrenticeHall, Englewood Cliffs, N.J., 1973. MR 0443377 (56:1747)
 [12]
 V. STREETER, Fluid Mechanics, 5th ed., McGrawHill, New York, 1971.
 [13]
 L. WAHLBIN, "A dissipative Galerkin method applied to some quasilinear hyperbolic equations," R.A.I.R.O. Anal. Numer., v. 8, 1974, pp. 109117. MR 0368447 (51:4688)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005834892
PII:
S 00255718(1980)05834892
Article copyright:
© Copyright 1980
American Mathematical Society
