A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations

Journal of Computational Physics

Volumn 223, Issue 1, 2007, Pages 398-415

  Cheng, Yingda ^a   Shu, Chi Wang ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Convex Hamiltonian; Discontinuous Galerkin; Hamilton Jacobi equations; High order accuracy

Indexed keywords

ENTROPY; FINITE ELEMENT METHOD; GALERKIN METHODS; NUMERICAL METHODS; PHYSICAL PROPERTIES; POLYNOMIALS;

CONSERVATION LAW; CONVEX HAMILTONIAN; DISCONTINOUS GALERKIN METHODS; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN FINITE-ELEMENT METHOD; ERROR ESTIMATES; HAMILTON - JACOBI EQUATIONS; HIGHER ORDER ACCURACY; SOURCE TERMS; STABILITY ESTIMATES;

HAMILTONIANS;

EID: 33947097945     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.09.012     Document Type: Article

References (19)

1. Cockburn B., Hou S., and Shu C.-W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Mathematics of Computation 54 (1990) 545-581
2. Cockburn B., Lin S.-Y., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems. Journal of Computational Physics 84 (1989) 90-113
3. Cockburn B., and Shu C.-W. The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Mathematical Modelling and Numerical Analysis 25 (1991) 337-361
4. Cockburn B., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Mathematics of Computation 52 (1989) 411-435
5. Cockburn B., and Shu C.-W. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics 141 (1998) 199-224
6. Crandall M., and Lions P.L. Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 277 (1983) 1-42
7. Gottlieb S., and Shu C.-W. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation 67 (1998) 73-85
8. Gottlieb S., Shu C.-W., and Tadmor E. Strong stability preserving high order time discretization methods. SIAM Review 43 (2001) 89-112
9. Hu C., and Shu C.-W. A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM Journal on Scientific Computing 21 (1999) 666-690
10. Jiang G., and Peng D. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM Journal on Scientific Computing 21 (1999) 2126-2143
11. Jiang G., and Shu C.-W. On cell entropy inequality for discontinuous Galerkin methods. Mathematics of Computation 62 (1994) 531-538
12. Lepsky O., Hu C., and Shu C.-W. The Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations. Applied Numerical Mathematics 33 (2000) 423-434
13. LeVeque R. High-resolution conservative algorithms for advection in incompressible flow. SIAM Journal on Numerical Analysis 33 (1996) 627-665
14. Li F., and Shu C.-W. Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Applied Mathematics Letters 18 (2005) 1204-1209
15. S. Osher, J. Yan, A new discontinuous Galerkin method for the Hamilton-Jacobi equation, in preparation.
16. Osher S., and Shu C.-W. High order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM Journal on Numerical Analysis 28 (1991) 907-922
17. Shu C.-W., and Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77 (1988) 439-471
18. Yan J., and Shu C.-W. A local discontinuous Galerkin method for KdV type equations. SIAM Journal on Numerical Analysis 40 (2002) 769-791
19. Zhang Y.-T., and Shu C.-W. High order WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM Journal on Scientific Computing 24 (2003) 1005-1030