On the convergence of a finite element method for a nonlinear hyperbolic conservation law
HTML articles powered by AMS MathViewer
- by Claes Johnson and Anders Szepessy PDF
- Math. Comp. 49 (1987), 427-444 Request permission
Abstract:
We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers’ equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function u, then u is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers’ equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.References
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724
- Amiram Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys. 49 (1983), no. 1, 151–164. MR 694161, DOI 10.1016/0021-9991(83)90118-3
- T. J. R. Hughes and T. E. Tezduyar, Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 217–284. MR 759810, DOI 10.1016/0045-7825(84)90157-9 T. J. Hughes, M. Mallet & L. P. Franca, "Entropy stable finite element methods for compressible fluids: Application to high Mach number flows with shocks," in Finite Elements for Nonlinear Problems (P. Bergen, K. J. Bathe and W. Wunderlich, eds.), Springer, Berlin, 1986, pp. 761-773.
- T. J. R. Hughes, L. P. Franca, and M. Mallet, A new finite element formulation for computational fluid dynamics. I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553, DOI 10.1016/0045-7825(86)90127-1
- T. J. R. Hughes, L. P. Franca, and M. Mallet, A new finite element formulation for computational fluid dynamics. I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553, DOI 10.1016/0045-7825(86)90127-1
- Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
- Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
- Claes Johnson, Uno Nävert, and Juhani Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 285–312. MR 759811, DOI 10.1016/0045-7825(84)90158-0
- Claes Johnson and Jukka Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp. 47 (1986), no. 175, 1–18. MR 842120, DOI 10.1090/S0025-5718-1986-0842120-4 C. Johnson, "Streamline diffusion methods for problems in fluid mechanics," in Finite Elements in Fluids, vol. 6 (Gallagher et al., eds.), Wiley, New York, 1985. C. Johnson & A. Szepessy, On the Convergence of Streamline Diffusion Finite Element Methods for Hyperbolic Conservation Laws, Numerical Methods for Compressible Flows—Finite Difference, Element and Volume Techniques—AMD vol. 78 (T. E. Tezduyar and T. J. R. Hughes, eds.).
- Jeffrey Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one, Comm. Math. Phys. 106 (1986), no. 3, 481–484. MR 859822
- Eitan Tadmor, Skew-selfadjoint form for systems of conservation laws, J. Math. Anal. Appl. 103 (1984), no. 2, 428–442. MR 762567, DOI 10.1016/0022-247X(84)90139-2
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 49 (1987), 427-444
- MSC: Primary 65M10; Secondary 65M60
- DOI: https://doi.org/10.1090/S0025-5718-1987-0906180-5
- MathSciNet review: 906180