High order two dimensional nonoscillatory methods for solving Hamilton-Jacobi scalar equations
The paper is commited to constructing high-order accurate numerical methods for solving nonlinear partial differential equations whose solutions (or their derivatives) develop steep gradients, here for the special case of the so-called nonlinear scalar Hamilton-Jacobi equations (of first order). A polynomial finite element technique, based on the ``essentially nonoscillatory schemes'' of the second author and \textit{J. A. Sethian} [J. Comput. Phys. 79, No. 1, 12-49 (1988; Zbl 0659.65132)], is used in an arbitrary domain, even with complicated boundaries (discretized in a conventional manner), combined with high-order Runge-Kutta schemes for the time steps. Numerical experiments are made for an optimal control problem, for a combustion problem, and for a problem due to P. L. Lions: a shape reconstruction from a shading problem, see also \textit{E. Rouy} and \textit{A. Tourin} [SIAM J. Number. Anal. 29, No. 3, 867-884 (1992; Zbl 0754.65069)].
nonlinear scalar Hamilton-Jacobi equations
scientific article; zbMATH DE number 881851