The first author and J. A. Sethian [J. Comput. Phys. 79, No. 1, 12- 49 (1988; Zbl 0659.65132)] constructed essentially nonoscillatory (ENO) schemes for the Hamilton-Jacobi equation and its perturbations, arising in front propagation problems.
In this paper a more general ENO scheme construction procedure is provided mainly by considering different multidimensional monotone building blocks. The schemes are numerically tested on a variety of one- and two-dimensional problems including a problem related to control optimization, checking the accuracy in smooth regions, resolution of discontinuities in derivatives, and the phenomenon of convergence to viscosity solutions.