Geodesic
On the numerical approximation of first-order Hamilton-Jacobi equations
International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 3, pp. 403-412.

Voir la notice de l'article provenant de la source Library of Science

Some methods for the numerical approximation of time-dependent and steady first-order Hamilton-Jacobi equations are reviewed. Most of the discussion focuses on conformal triangular-type meshes, but we show how to extend this to the most general meshes. We review some first-order monotone schemes and also high-order ones specially dedicated to steady problems.
Keywords: approximation of Hamilton-Jacobi equations, viscous solution, Cauchy-Dirichlet problem, triangular mesh
Mots-clés : równanie Hamiltona-Jacobiego, aproksymacja, zagadnienie Cauchy'ego-Dirichleta, siatka trójkątna 
@article{IJAMCS_2007_17_3_a9,
     author = {Abgrall, R. and Perrier, V.},
     title = {On the numerical approximation of first-order {Hamilton-Jacobi} equations},
     journal = {International Journal of Applied Mathematics and Computer Science},
     pages = {403--412},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a9/}
}
TY  - JOUR
AU  - Abgrall, R.
AU  - Perrier, V.
TI  - On the numerical approximation of first-order Hamilton-Jacobi equations
JO  - International Journal of Applied Mathematics and Computer Science
PY  - 2007
SP  - 403
EP  - 412
VL  - 17
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a9/
LA  - en
ID  - IJAMCS_2007_17_3_a9
ER  - 
%0 Journal Article
%A Abgrall, R.
%A Perrier, V.
%T On the numerical approximation of first-order Hamilton-Jacobi equations
%J International Journal of Applied Mathematics and Computer Science
%D 2007
%P 403-412
%V 17
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a9/
%G en
%F IJAMCS_2007_17_3_a9
Abgrall, R.; Perrier, V. On the numerical approximation of first-order Hamilton-Jacobi equations. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 3, pp. 403-412. http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a9/

[1] Abgrall R. (1996): Numerical Discretization of First Order Hamilton-Jacobi Equations on Triangular Meshes. Communications on Pure and Applied Mathematics, Vol. XLIX, No. 12, pp. 1339-1373.

[2] Abgrall R. (2004): Numerical discretization of boundary conditions for first order Hamilton Jacobi equations. SIAM Journal on Numerical Analyis, Vol. 41, No. 6, pp. 2233-2261.

[3] Abgrall R. (2007): Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations. (in revision).

[4] Abgrall R. and Perrier V. (2007): Error estimates for Hamilton-Jacobi equations with boundary conditions. (in preparation).

[5] Augoula S. and Abgrall R. (2000): High order numerical discretization for Hamilton-Jacobi equations on triangular meshes. Journal of Scientific Computing, Vol. 15, No. 2, pp. 197-229.

[6] Bardi M. and Evans L.C. (1984): On Hopf's formula for solutions of first order Hamilton-Jacobi equations. Nonlinear Analysis Theory: Methods and Applications, Vol. 8, No. 11, pp. 1373-1381.

[7] BardiM. and Osher S. (1991): The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations. SIAM Journal onMathematical Analysis, Vol. 22, No. 2, pp. 344-351.

[8] Barles G. (1994): Solutions de viscosité des équations de Hamilton-Jacobi. Paris: Springer.

[9] Barles G. and Souganidis P.E. (1991): Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis, Vol. 4, No. 3, pp. 271-283.

[10] Crandall M.G. and Lions P.L. (1984): Two approximations of solutions of Hamilton-Jacobi equations. Mathematics of Computation, Vol. 43, No. 167, pp. 1-19.

[11] Deckelnick K. and Elliot C.M. (2004): Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities. Interfaces and Boundary, Vol. 6, No. 3, pp. 329-349.

[12] Hu C. and Shu C.W. (1999): A discontinuous Galerkin finite element method for Hamilton Jacobi equations. SIAMJournal on Scientific Computing, Vol. 21, No. 2, pp. 666-690.

[13] Li F. and Shu C.W. (2005): Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Applied Mathematics Letters, Vol. 18, No. 11, pp. 1204-1209.

[14] Lions P.-L. (1982): Generalized Solutions of Hamilton-Jacobi Equations. Boston: Pitman.

[15] Osher S. and Shu C.W. (1991): High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM Journal on Numerical Analysis, Vol. 28, No. 4, pp. 907-922.

[16] Qiu J. and Shu C.W. (2005): Hermite WENO schemes for Hamilton-Jacobi equations. Journal of Computational Physics, Vol. 204, No. 1, pp. 82-99.

[17] Zhang Y.T. and Shu C.W. (2003): High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM Journal on Scientific Computing, Vol. 24, No. 3, pp. 1005-1030.