What is the simplest way to flush cache at a predetermined date and time?
May be How can I schedule a rule to always be run at a specific hour of the day? can be useful , but using cron, build a module and in it's hook_cron function check for the current time, if it is your specific time use the cache_clear_all function and it's parameters to clear the page and else cache. just make sure that the cron is ran on Wednesday day at midnight. it's periodic run time should come to 24:59.

you can also check the clear drupal cache - php script for more options of clearing the cache statement.