Joy,

You did not give much information such as the sort of water temperature control your engine has, whether it is a new problem or not, etc. More details will help diagnose it. However, the symptoms are identical to ones I experienced. On my older model A4 the hot water exits the engine to a tee that contains the thermostat. One side of the tee goes to a bypass line that returns water to the inlet to the water pump when the engine is cold. As the engine warms up, the thermostat opens and allows more of the water out the other side of the tee to the exhaust and restricts the bypass. After my engine was rebuilt, the yard (not the rebuilder) installed the tee backwards so that when cold there was no bypass and everything looked fine, except that the engine warmed up slowly. When it did warm up, the thermostat opened but now forced most of the water to the bypass and caused rapid overheating.

John