Now that many don’t go to the mikvaos, is it permissible to purify oneself on Shabbos with tisha kavim on Shabbos in the shower?
It is permissible, however it has to be with only cold water, as turning on the hot water is causing new water to cook on Shabbos. You must be very careful though to make sure that you don’t add mistakenly add hot water to the shower, because turning on the hot water causes new water to cook.
Minchas Yitzchok 9-34, Nishmas Shabbos 2 pg. 219. See Yesodei Yeshurun pg. 400, who suggests placing something on the hot water faucet as a reminder not to turn the hot water .