A Weighted Essentially Nonoscillatory, Large Time-Step Scheme for Hamilton-Jacobi Equations

We investigate the application of weighted essentially nonoscillatory (WENO) reconstructions to a class of semi-Lagrangian schemes for first order time-dependent Hamilton--Jacobi equations. In particular, we derive a general form of the scheme, study sufficient conditions for its convergence with high-order reconstructions, and perform numerical tests to study its efficiency. In addition, we prove that the weights of the WENO interpolants are positive for any order.

[1]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[2]  R. Courant,et al.  On the solution of nonlinear hyperbolic differential equations by finite differences , 1952 .

[3]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[4]  R. Abgrall.  Numerical discretization of the first‐order Hamilton‐Jacobi equation on triangular meshes , 1996 .

[5]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[6]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[7]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[8]  J. Strain.  Semi-Lagrangian Methods for Level Set Equations , 1999 .

[9]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[10]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[11]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[12]  S. SIAMJ.,et al.  CENTRAL SCHEMES FOR MULTIDIMENSIONAL HAMILTON – JACOBI EQUATIONS , 2003 .

[13]  Wang Hai-bing,et al.  High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations , 2006 .

[14]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[15]  P. Lions.  Optimal control and viscosity solutions , 1985 .

[16]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[17]  Chi-Wang Shu,et al.  High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes , 2003, SIAM J. Sci. Comput..

[18]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[19]  Marizio Falcone,et al.  Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations , 1994 .

[20]  M. Falcone,et al.  Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods , 2002 .

[21]  Roberto Ferretti,et al.  Convergence of Semi-Lagrangian Approximations to Convex Hamilton-Jacobi Equations under (Very) Large Courant Numbers , 2002, SIAM J. Numer. Anal..

[22]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[23]  Steve Bryson,et al.  Central Schemes for Multidimensional Hamilton-Jacobi Equations , 2003, SIAM J. Sci. Comput..

[24]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[25]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[26]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[27]  Chi-Tien Lin,et al.  High-Resolution Nonoscillatory Central Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[28]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[29]  Chi-Wang Shu,et al.  A technique of treating negative weights in WENO schemes , 2000 .

[30]  M. Falcone,et al.  An Approximation Scheme for Evolutive Hamilton-Jacobi Equations , 1999 .