The disc is a Memorex DVD+R. It came half used and I would like it to be erased. It can be a physical procedure, if necessary. Maybe there is a way to take up all space and then virtually use it as empty?
If you have recorded the disk, and if you have not finalized it, it is possible to add more data, but not to erase it, AFAIK. However, if you received the disk from someone else and are able to read the data, then it has been finalized.
Finalization* puts support data, such as a permanent table of contents on the disk, rather than relying on a temporary one stored on the machine writing data to the disk.
* Apparently, DVD+VR does not require finalization.
In all seriousness, the best solution for most peoples needs is shredding. You can get a shredder pretty cheap at most any office supply store (most standard home/office shredders will do CD/DVDs). That would be an easy, reliable solution with reasonable security.
Secondly, if you are not too environmentally conscious, applying heat to the disk will corrupt and destroy the data. It's a pain to do, and is woefully irresponsible, but it provides a high degree of certainty that the data on a melted disk is unrecoverable against even the most powerful adversaries.
Scratching it up and breaking it is sufficient for most use-cases. It's not as convenient as a shredder, but its cheap and easy enough.