Car cut off when stopped on 2006 Chevrolet Cobalt

My car shuts off when i stop at a light, or sign, etc. When i start it back up, the gas needle drops, and slowly goes back up. But this all occur when my gas is under half a tank. When it's full, there is no problem at all. Also my rpm needle goes up and down, and my car stutters. What is causing this?

1 answer
possible fuel filter, fuel injectors, or water might be inside the tank