A finite element method for first-order hyperbolic systems

Author:
Mitchell Luskin

Journal:
Math. Comp. **35** (1980), 1093-1112

MSC:
Primary 65N30; Secondary 65M15, 76N15

MathSciNet review:
583489

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new finite element method is proposed for the numerical solution of a class of initial-boundary value problems for first-order 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 two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**0371077**, 10.1090/S0025-5718-1975-0371077-0**[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****[3]**Todd Dupont,*Galerkin methods for first order hyperbolics: an example*, SIAM J. Numer. Anal.**10**(1973), 890–899. MR**0349046****[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**, 10.1137/0716032**[6]**Milton Lees,*A linear three-level difference scheme for quasilinear parabolic equations*, Math. Comp.**20**(1966), 516–522. MR**0207224**, 10.1090/S0025-5718-1966-0207224-5**[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**, 10.1016/0022-247X(81)90192-X**[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 finite-element barotropic model,"*J. Phys. Oceanogr.*, v. 8, 1979, pp. 323-343.**[10]**H. H. Rachford Jr.,*Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations*, SIAM J. Numer. Anal.**10**(1973), 1010–1026. MR**0339519****[11]**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377****[12]**V. STREETER,*Fluid Mechanics*, 5th ed., McGraw-Hill, 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. R-2, 109–117 (English, with Loose French summary). MR**0368447**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
65M15,
76N15

Retrieve articles in all journals with MSC: 65N30, 65M15, 76N15

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1980-0583489-2

Article copyright:
© Copyright 1980
American Mathematical Society