TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework

Authors:
Bernardo Cockburn and Chi-Wang Shu

Journal:
Math. Comp. **52** (1989), 411-435

MSC:
Primary 65M60; Secondary 35L65, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983311-4

MathSciNet review:
983311

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This is the second 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 . In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

**[1]**A. Bourgeat & B. Cockburn,*The TVD-projection method for solving implicit numerical schemes for scalar conservation laws*:*A numerical study of a simple case*, IMA Preprint Series #311, University of Minnesota, April 1987. To appear in*SIAM J. Sci. Statist. Comput.*, March 1989. MR**982223 (90g:65116)****[2]**G. Chavent &, B. Cockburn,*The local projection*-*discontinuous-Galerkin finite element method for scalar conservation laws*, IMA Preprint Series #341, University of Minnesota, September 1987, to appear in .**[3]**G. Chavent & G. Salzano, "A finite element method for the ID water flooding problem with gravity,"*J. Comput. Phys.*, v. 45, 1982, pp. 307-344. MR**666166 (83g:76089)****[4]**B. Cockburn,*The quasi-monotone schemes for scalar conservation laws*, I, II,*and*III, IMA Preprint Series #263, 268, 277, University of Minnesota, September and October 1986.**[5]**B. Cockburn & C.-W. Shu,*The Runge-Kutta local projection*-*discontinuous-Galerkin finite element method for scalar conservation laws*, IMA Preprint Series #388, University of Minnesota, 1988.**[6]**M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws,"*Math. Comp.*, v. 34, 1980, pp. 1-21. MR**551288 (81b:65079)****[7]**A. Harten, "On a class of high resolution total-variation-stable finite-difference schemes,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 1-23. MR**731210 (85f:65085)****[8]**A. Harten,*Preliminary results on the extension of ENO schemes to two-dimensional problems*, Proc. Internat. Conf. on Hyperbolic Problems, Saint-Etienne, January 1986. MR**910102 (88k:65085)****[9]**A. Harten & S. Osher, "Uniformly high-order accurate nonoscillatory schemes, I,"*SIAM J. Numer. Anal.*, v. 24, 1987, pp. 279-309. MR**881365 (90a:65198)****[10]**A. Harten, B. Engquist, S. Osher & S. Chakravarthy, "Uniformly high order accurate essentially non-oscillatory schemes, III,"*J. Comput. Phys.*, v. 71, 1987, pp. 231-303. MR**897244 (90a:65199)****[11]**T. Hughes & A. Brook, "Streamline upwind-Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,"*Comput. Methods Appl. Mech. Engrg.*, v. 32, 1982, pp. 199-259. MR**679322 (83k:76005)****[12]**T. Hughes, L. P. Franca, M. Mallet & A. Misukami, "A new finite element formulation for computational fluid dynamics, I, II, III and IV,"*Comput. Methods Appl. Mech. Engrg.*, v. 54, 58, 1986, pp. 223-234, 341-355; pp. 305-328, 329-336. MR**831553 (87f:76010a)****[13]**C. Johnson & J. Pitkäranta, "An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation,"*Math. Comp.*, v. 46, 1986, pp. 1-26. MR**815828 (88b:65109)****[14]**C. Johnson & J. Saranen, "Streamline diffusion methods for problems in fluid mechanics,"*Math. Comp.*, v. 47, 1986, pp. 1-18. MR**842120 (88b:65133)****[15]**C. Johnson & A. Szepessy, "On the convergence of a finite element method for a nonlinear hyperbolic conservation law,"*Math. Comp.*, v. 49, 1987, pp. 427-444. MR**906180 (88h:65164)****[16]**C. Johnson, A. Szepessy & P. Hansbo, "On the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws," preprint. MR**995210 (90j:65118)****[17]**B. Van Leer, "Towards the ultimate conservation difference scheme, II and V,"*J. Comput. Phys.*, v. 14 and 32, 1974 and 1979, pp. 361-376 and 101-136.**[18]**P. Lesaint & P.-A. Raviart, "On a finite element method for solving the neutron transport equation," in*Mathematical Aspects of Finite Elements in Partial Differential Equations*(C. de Boor, ed.), Academic Press, 1974, pp. 89-145. MR**0658142 (58:31918)****[19]**K. W. Morton & P. K. Sweby, "A comparison of flux limited difference methods and characteristic Galerkin methods for shock modelling,"*J. Comput. Phys.*, v. 73, 1987, pp. 203-230.**[20]**S. Osher, "Convergence of generalized MUSCL schemes,"*SIAM J. Numer. Anal.*, v. 22, 1985, pp. 947-961. MR**799122 (87b:65147)****[21]**S. Osher & S. Chakravarthy, "High resolution schemes and the entropy condition,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 955-984. MR**760626 (86a:65086)****[22]**S. Osher & E. Tadmor, "On the convergence of difference approximations to scalar conservation laws,"*Math. Comp.*, v. 50, 1988, pp. 19-51. MR**917817 (89m:65086)****[23]**R. Sanders, "A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws,"*Math. Comp.*, v. 51, 1988, pp. 535-558. MR**935073 (89c:65104)****[24]**C.-W. Shu, "TVB uniformly high-order schemes for conservation laws,"*Math. Comp.*, v. 49, 1987, pp. 105-121. MR**890256 (89b:65208)****[25]**C.-W. Shu, "TVB boundary treatment for numerical solutions of conservation laws,"*Math. Comp.*, v. 49, 1987, pp. 123-134. MR**890257 (88h:65179)****[26]**C.-W. Shu, "Total-Variation-Diminishing time discretizations,"*SIAM J. Sci. Statist. Comput.*, v. 9, 1988, pp. 1073-1084. MR**963855 (90a:65196)****[27]**C.-W. Shu & S. Osher, "Efficient implementation of essentially non-oscillatory shock-capturing schemes,"*J. Comput. Phys.*, v. 77, 1988, pp. 439-471. MR**954915 (89g:65113)****[28]**P. Sweby, "High resolution schemes using flux limiters for hyperbolic conservation laws,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 995-1011. MR**760628 (85m:65085)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M60,
35L65,
65N30

Retrieve articles in all journals with MSC: 65M60, 35L65, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983311-4

Keywords:
TVD,
TVB,
Runge-Kutta,
discontinuous finite elements,
conservation law

Article copyright:
© Copyright 1989
American Mathematical Society