Preserving positivity for hyperbolic PDEs using variable-order finite elements with bounded polynomials

Applied Numerical Mathematics

Volumn 52, Issue 2-3 SPEC. ISS., 2005, Pages 197-217

  Berzins, M ^a  

^a Department of Electrical and Computer Engineering   (United States)

Author keywords

Bounded polynomials; Finite elements; Hyperbolic equations; Positivity

Indexed keywords

COMPUTATIONAL METHODS; GALERKIN METHODS; MATRIX ALGEBRA; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING;

BOUNDED POLYNOMIALS; FINITE ELEMENTS; HYPERBOLIC EQUATIONS; POSITIVITY;

FINITE ELEMENT METHOD;

EID: 10444225545     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2004.08.031     Document Type: Conference Paper

References (24)

1. F. Arandiga, G. Chiavassa, R. Donat, Harten's Framework for Multiresolaution with Applications from Conservation Laws to Image Compression, GrAN Report 01-01, May 15th 2001, Departament de Matematica Aplicada, Universitat de Valencia, Valencia, Spain.
2. M. Berzins, Modified mass matrices and positivity preservation for hyperbolic and parabolic PDEs, Comm. Numer. Methods Engrg. 17 (2001) 659-666.
3. M. Berzins, Variable-order finite elements and positivity preservation for hyperbolic PDEs, Appl. Numer. Math. 48 (2004) 271-292.
4. V.S. Borisov, S. Sorek, On monotonicity of difference schemes for computational physics, SIAM J. Sci. Comput. 25 (2004) 1557-1584.
5. K.E. Brenan, S.L. Campbell, L.R. Petzold, Numerical Solution of Initial-Value problems in Differential Algebraic Equations, North-Holland, Amsterdam, 1989.
6. R. Biswas, K. Devine, J.E. Flaherty, Parallel adaptive finite element methods for conservation laws, Appl. Numer. Math. 14 (1995) 2557-2583.
7. B. Cockburn, C.-W. Shu, Nonlinearly stable compact schemes for shock calculations, SIAM J. Numer. Anal. 31 (1994) 607-627.
8. B. Cockburn, G.E. Karniadakis, C.-W. Shu (Eds.), Discontinuous Galerkin Methods, Theory Computations and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000, pp. 3-53.
9. S.K. Godunov, Finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Math. Sbornik 47 (1959) 271-306.
10. D. Gottlieb, C.-W. Shu, Resolution properties of the Fourier method for discontinuous waves, in: C. Bernardi, Y. Maday (Eds.), Analysis Algorithms and Applications of Spectral and High-Order Methods for Partial Differential Equations, Selected Papers from the international Conference on Spectral and High Order Methods (ICOSAHOM'92) Le Corum, Montpellier, France, June 22-26 1992, North-Holland, Amsterdam, 1994, p. 27.
11. D. Gottlieb, J.S. Hesthaven, Spectral methods for hyperbolic problems, J. Comput. Appl. Math. 128 (2001) 83-131.
12. P.M. Gresho, R.L. Saini, Incompressible Flow and the Finite Element Method, Advection Diffusion, vol. 1, Wiley, 1998.
13. A. Harten, G. Zwas, Self adjusting hybrid schemes for shock computations, J. Comput. Phys. 9 (1972) 568-583.
14. A. Harten, Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Comm. Pure and Appl. Math. XIVIII (1995) 1305-1342.
15. F.B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956.
16. T.Y. Hou, P.G. Le Floch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comput. 62 (1994) 497-530.
17. M.E. Hubbard, M. Berzins, A positivity preserving finite element method for hyperbolic partial differential equations, in: S. Armfield, P. Morgan, K. Srinivas (Eds.), Computational Fluid Dynamics, Springer, Berlin, 2003, pp. 205-210.
18. C. Laney, Computational Gasdynamics, Cambridge University Press, Cambridge, 1998.
19. M.G. Larson, T.J.R. Hughes, G. Engel, L. Mazzei, The continuous Galerkin method is conservative, J. Comput. Phys. 163 (2) (2000) 467-488.
20. A. Segal, Finite element methods for advection-diffusion equations, in: C.B. Vreugdenhil, B. Koren (Eds.), Numerical Methods for Advection Diffusion Problems, Notes on Numerical Fluid Mechanics, vol. 45, Vieweg, Braunschweig, 1993, pp. 195-214.
21. C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77 (1988) 439-471.
22. C. Schwab, p- and hp- Finite Element Methods-Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation, Clarendon, Clarendon, TX, 1998.
23. T.W.h. Sheu, S.F. Tsai, M.M.T. Wang, A monotone finite element method with test space of Legendre polynomials, Comput. Methods Appl. Mech. Engrg. 143 (1997) 349-372.
24. T.W.h. Sheu, P.H. Lee, R.K. Lin, Development of a high resolution hyperbolic model on quadratic elements, Comput. Methods Appl. Mech. Engrg. 192 (2003) 5036-5037.