Nondominated sorting is a fundamental combinatorial problem in multiobjective optimization and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science. In a previous work [SIAM J. Math. Anal., 46 (2014), pp. 603--638], we showed that nondominated sorting has a continuum limit that corresponds to solving a Hamilton--Jacobi equation. In this work we present and analyze a fast numerical scheme for this Hamilton--Jacobi equation and show how it can be used to design a fast algorithm for approximate nondominated sorting.