This is an interesting promotion of an applied math result. From their promotional material it looks promising, though the unusual promotional approach makes me worry.

The actual paper is at https://www.sciencedirect.com/science/article/pii/S016653161.... This is a pretty obscure journal. The paper is pretty "soft" -- lots of numerical testing of their approach vs. other well-known approaches and not very much theoretical analysis of convergence rates or such.

The main claim seems to be that their approach has better numerical properties for discontinuous functions and that it can be effectively implemented to high order using double precision arithmetic.