Unless you absolutely need to update firmware leave your rewriter as it is - you only need one small error and the drive will almost certainly be only fit for the dustbin.
However, if you still wish to go ahead, the Philips DVDRW8631 is apparently the same drive as the Benq DW1620.
click here (see note 5-1-6).
"First the Philips (unzipped) version Gd30
( click here ); if this doesn’t work, try the BenQ version B7W9 (unzipped) version ( click here )
The general opinion on other forums is that it's not a particularly good rewriter.
But a new one, if it all goes wrong, will be less than £20 for one of the latest top brand DVD rewriters from Scan, Dabs etc.