High-Order Essentially Nonoscillatory Schemes for Hamilton–Jacobi Equations

Abstract

Hamilton–Jacobi (H–J) equations are frequently encountered in applications, e.g., in control theory and differential games. H–J equations are closely related to hyperbolic conservation laws—in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations. In this paper high-order essentially nonoscillatory (ENO) schemes for H–J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed.