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Numerical schemes and rates of convergence for the Hamilton-Jacobi equation continuum limit of nondominated sorting

Nondominated sorting arranges a set of points in Euclidean space into layers by repeatedly removing the coordinatewise minimal elements. It was recently shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. The obvious numerical scheme for this PDE has a slow convergence rate of O(h^1/n) for a grid of spacing h>0 in dimension n. In this paper, we introduce two new numerical schemes that have formal rates of O(h) and we prove the usual O(h^1/2) theoretical rates. We also present the results of numerical simulations illustrating the difference between the formal and theoretical rates.