Holt winters with daily and weekly period

We have data that has a clear daily periodicity (spike in the morning, peak at 10am that gradually tails off) as well as a clear weekly periodicity (Monday has higher peaks then Tuesday, Tuesday higher than Wednesday etc, saturday and sunday are very low). We have tried to use holt winters to predict these values but haven’t gotten a great fit. I believe this is because it is assuming all days are the same rather then having a pattern within a pattern.

Does anyone have any experience with other algorithms that support this kind of cyclical data?

This presentation comes to mind, where the Facebook Prophet algorithm is used to model the seasonal patterns in your data. https://www.youtube.com/watch?v=raEyZEryC0k

I have used Prophet with some partial success to predict traffic patterns. It predicts reasonably well if the daily pattern is consistent (not on point, but would be enough as an approximation). However, if the pattern changes significantly in the weekends (which is typical for some situations, where user’s only consume services during regular week days), Prophet will not be able to pick on that very well. Truth be told, I did not spend a lot of time trying to tune the parameters, so there may be room for improvement.