# On the numerical approximation of first-order Hamilton-Jacobi equations

International Journal of Applied Mathematics and Computer Science (2007)

- Volume: 17, Issue: 3, page 403-412
- ISSN: 1641-876X

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topAbgrall, Rémi, and Perrier, Vincent. "On the numerical approximation of first-order Hamilton-Jacobi equations." International Journal of Applied Mathematics and Computer Science 17.3 (2007): 403-412. <http://eudml.org/doc/207846>.

@article{Abgrall2007,

abstract = {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.},

author = {Abgrall, Rémi, Perrier, Vincent},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {Cauchy-Dirichlet problem; viscous solution; triangular mesh; approximation of Hamilton-Jacobi equations; discontinuous Galerkin method; Godunov scheme; numerical examples; Dirichlet problem; Cauchy problem},

language = {eng},

number = {3},

pages = {403-412},

title = {On the numerical approximation of first-order Hamilton-Jacobi equations},

url = {http://eudml.org/doc/207846},

volume = {17},

year = {2007},

}

TY - JOUR

AU - Abgrall, Rémi

AU - Perrier, Vincent

TI - On the numerical approximation of first-order Hamilton-Jacobi equations

JO - International Journal of Applied Mathematics and Computer Science

PY - 2007

VL - 17

IS - 3

SP - 403

EP - 412

AB - 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.

LA - eng

KW - Cauchy-Dirichlet problem; viscous solution; triangular mesh; approximation of Hamilton-Jacobi equations; discontinuous Galerkin method; Godunov scheme; numerical examples; Dirichlet problem; Cauchy problem

UR - http://eudml.org/doc/207846

ER -

## References

top- 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. Zbl0870.65116
- 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. Zbl1058.65102
- Abgrall R. (2007): Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations. (in revision). Zbl1197.65167
- Abgrall R. and Perrier V. (2007): Error estimates for Hamilton-Jacobi equations with boundary conditions. (in preparation). Zbl1147.65323
- 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. Zbl1077.65506
- 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. Zbl0569.35011
- Bardi M. and Osher S. (1991): The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations. SIAM Journal on Mathematical Analysis, Vol.22, No.2, pp.344-351. Zbl0733.35073
- Barles G. (1994): Solutions de viscosé des équations de Hamilton-Jacobi. Paris: Springer.
- 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. Zbl0729.65077
- 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. Zbl0556.65076
- Deckelnick K. and Elliot C.M. (2004): Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuies. Interfaces and Boundary, Vol.6, No.3, pp.329-349. Zbl1081.35115
- Hu C. and Shu C.W. (1999): A discontinuous Galerkin finite element method for Hamilton Jacobi equations. SIAM Journal on Scientific Computing, Vol.21, No.2, pp.666-690. Zbl0946.65090
- 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. Zbl1118.65105
- Lions P.-L. (1982): Generalized Solutions of Hamilton-Jacobi Equations. Boston: Pitman.
- 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. Zbl0736.65066
- 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. Zbl1070.65078
- 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 Zbl1034.65051

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