I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or computer algebra. Mathematica is now running for 10 days without answer.

A similar question description here was asked a couple of years ago but because there was no satisfying answer I raise this question again. I would appreciate if anyone could tell help me with the following two questions

- if there is any hope to compute the Groebner basis?
- if so are there fast at best parallel ready to use programms for computing the Groebner basis?

Any answers, comments and reference are appreciated.