TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: One-dimensional systems. (English) Zbl 0677.65093

Summary: [For part II see the first and the third author, Math. Comput. 52, No.186, 411-435 (1989; Zbl 0662.65083).]
This is the third paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws \(u_ t+\sum^{d}_{i=1}(f_ i(u))_{x_ i}=0.\) In this paper we present the method in a system of equations, stressing the point of how to use the weak form in the component spaces, but to use the local projection limiting in the characteristic fields, and how to implement boundary conditions. A 1-dimensional system is thus chosen as a model. Different implementation techniques are discussed, theories analogous to scalar cases are proven for linear systems, and numerical results are given illustrating the method on nonlinear systems. Discussions of handling complicated geometries via adaptive triangle elements will appear in future papers.


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35L65 Hyperbolic conservation laws


Zbl 0662.65083
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[1] {\scG. Chavent and B. Cockburn}, IMA Preprint Series No. 341, University of Minnesota, September 1987; M^{2}AN, in press.
[2] Chavent, G.; Salzano, G., J. comput. phys., 45, 307, (1982)
[3] {\scB. Cockburn and C.-W. Shu}, IMA Preprint Series No. 388, University of Minnesota, January 1988;
[4] {\scB. Cockburn and C.-W. Shu}, IMA Preprint Series No. 392, University of Minnesota, March 1988;
[5] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., J. comput. phys., 71, 231, (1987)
[6] Harten, A.; Harten, A., (), J. comput. phys., 83, 148, (1989)
[7] Hughes, T.; Mallet, M., Finite element fluids, 6, 339, (1985)
[8] Johnson, C.; Pitkaranta, J., Math. comput., 46, 1, (1986)
[9] Lax, P., Commun. pure appl. math., 7, 159, (1954)
[10] Lesaint, P.; Raviart, P., Mathematical aspects of finite element in partial differential equations, (), 89
[11] Osher, S., SIAM J. num. anal., 22, 947, (1985)
[12] Roe, P., J. comput. phys., 43, 357, (1981)
[13] Shu, C.-W., Math. comput., 49, 105, (1987)
[14] Shu, C.-W., Math. comput., 49, 123, (1987)
[15] Shu, C.-W.; Osher, S., J. comput. phys., 77, 439, (1988)
[16] Shu, C.-W.; Osher, S.; Shu, C.-W.; Osher, S., (), J comput. phys., 83, 32, (1989)
[17] Sod, G., J. comput. phys., 27, 1, (1978)
[18] Woodward, P.; Colella, P., J. comput. phys., 54, 115, (1984)
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