Where should adverbs be placed?

There are two sentences:

I completely understand.
I understand completely.

Which one is correct and why?

Another example:

I slowly opened the door.
I opened the door slowly.

Solution 1:

There are several types of adverbs, and the rules concerning their position in a sentence vary. Here is a quick and useful summary.

Your second example includes the commonly used 'adverb of manner', which would normally be placed immediately following the object, if there is one, otherwise following the verb. So using this rule the 'correct' version would be:

I opened the door slowly.

Another example would be:

I ate my dinner slowly.

You could omit the object and it becomes:

I ate slowly.

In your first example I would also deem 'completely' to be an adverb of manner, so the correct usage would be:

I understand completely.

If, however, instead of 'completely' you had used 'occasionally' or 'rarely' then it would be an adverb of frequency, in which case the adverb moves before the verb and the sentence becomes:

I rarely understand.

Having said all that, I don't think there is ambiguity in any of the four examples you provided.

Solution 2:

They seem equally correct in your examples.

The order would give a slight difference in emphasis, though I bet people would disagree on which one highlights 'slowly' more in your second example.