People have done DNS with finite volume methods, which are 2nd-order accurate (correct me if I'm wrong). It will depend on many things, how large is the grid? how long do you need to run it? How scalable is your code on massively parallel machines?

In short, yes, you can do it. In general, higher-order methods are more efficient for large runs (will require less computational cost for the same level of accuracy). So it all comes down to whether you can afford to run the problem or not.