Abstract
In this paper, we are interested in some front propagation problems coming from control problems in d-dimensional spaces, with d≥2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an Hamilton-Jacobi-Bellman equation with discontinuous data, discretized by means of the antidissipative Ultra Bee scheme.
We develop an efficient dynamic storage technique suitable for handling front evolutions in large dimension. Then we propose a fast algorithm, showing its relevance on several challenging tests in dimension d=2,3,4. We also compare our method with the techniques usually used in level set methods. Our approach leads to a computational cost as well as a memory allocation scaling as O(N nb ) in most situations, where N nb is the number of grid nodes around the front. Moreover, we show on several examples the accuracy of our approach when compared with level set methods.
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References
Abgrall, R.: Numerical discretization of first-order Hamilton-Jacobi equation on triangular meshes. Commun. Pure Appl. Math. 49, 1339–1373 (1996)
Abgrall, R., Augoula, S.: High order numerical discretization for Hamilton-Jacobi equations on triangular meshes. J. Sci. Comput. 15, 197–229 (2000)
Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. J. Comput. Phys. 118, 269–277 (1995)
Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997)
Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., Van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994)
Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15, 1713–1742 (1990)
Bokanowski, O., Cristiani, E., Laurent-Varin, J., Zidani, H.: Hamilton-Jacobi-Bellman approach for the climbing problem for heavy launchers. Preprint (2009)
Bokanowski, O., Forcadel, N., Zidani, H.: Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous initial data. To appear in Math. Comput.
Bokanowski, O., Martin, S., Munos, R., Zidani, H.: An anti-diffusive scheme for viability problems. Appl. Numer. Math. 56, 1135–1254 (2006)
Bokanowski, O., Megdich, N., Zidani, H.: An adaptative antidissipative method for optimal control problems. Arima 5, 256–271 (2006)
Bokanowski, O., Megdich, N., Zidani, H.: Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous initial data. Numer. Math. (2009). doi:10.1007/s00211-009-0271-1
Bokanowski, O., Zidani, H.: Anti-diffusive schemes for linear advection and application to Hamilton-Jacobi-Bellman equations. J. Sci. Comput. 30, 1–33 (2007)
Crandall, M.G., Lions, P.-L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43, 1–19 (1984)
Desprès, B., Lagoutière, F.: Un schéma non linéaire anti-dissipatif pour l’équation d’advection linéaire. A non-linear anti-diffusive scheme for the linear advection equation. C. R. Acad. Sci. Paris, Sér. I, Math. 328, 939–944 (1999)
Desprès, B., Lagoutière, F.: Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. J. Sci. Comput. 16, 479–524 (2001)
Frankowska, H.: Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31, 257–272 (1993)
Hartmann, D., Meinke, M., Schroeder, W.: Differential equation based constrained reinitialization for level set methods. J. Comput. Phys. 227, 6821–6845 (2008)
Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)
Lagoutière, F.: A non-dissipative entropic scheme for convex scalar equations via discontinuous cell-reconstruction. C. R. Math. Acad. Sci. Paris 338, 549–554 (2004)
Lagoutière, F.: Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. thesis, University of Paris VI, Paris, France (2000)
Lin, C.-Y., Chung, Y.-C.: Efficient data compression methods for multidimensional sparse array operations based on the EKMR scheme. IEEE Trans. Comput. 52, 1640–1646 (2003)
Megdich, N.: Méthodes anti-dissipatives pour les equations de Hamilton-Jacobi-Bellman. Ph.D. thesis, University of Paris VI, Paris, France (2008)
Mitchell, I., Bayen, A., Tomlin, C.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Automat. Contr. 50, 947–957 (2005)
Osher, S.: A level set formulation for the solution of the Hamilton-Jacobi equations. SIAM J. Math. Anal. 24, 1145–1152 (1993)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)
Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28, 907–922 (1991)
Peng, D.P., Merriman, B., Osher, S., Zhao, H.K., Kang, M.J.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999)
Robins, G.: Robs algorithm. Appl. Math. Comput. 189, 314–325 (2007)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)
Shu, C.-W.: High order ENO and WENO schemes for computational fluid dynamics. In: High-order Methods for Computational Physics. Lect. Notes Comput. Sci. Eng., vol. 9, pp. 439–582. Springer, Berlin (1999)
Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible 2-phase flow. J. Comput. Phys. 114, 146–159 (1994)
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Bokanowski, O., Cristiani, E. & Zidani, H. An Efficient Data Structure and Accurate Scheme to Solve Front Propagation Problems. J Sci Comput 42, 251–273 (2010). https://doi.org/10.1007/s10915-009-9329-6
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DOI: https://doi.org/10.1007/s10915-009-9329-6