Q: In my neighborhood we fenced in about 50’x100′ for a dog park. The property we fenced off goes slightly downhill with a bowl in the middle which catches water and has become a mud pit. Most of the centipede grass that covered this yard has been destroyed by the dogs. Can you suggest any landscaping solutions to our problems?

A: One man has some good comments at 8 Tips for Landscaping with Dogs.
It sounds like you have too many dogs for the park to accommodate and be able to repair the damage. Bermudagrass is the fastest to recover from dog damage – but only in summer. Unless you limit the number of dogs using the park, your best surface may be decomposed granite (granite dust, granite sand).
Here are some more good thoughts on dog park design: Best Dog Park in the World.