Does the universe have a center away from which everything is accelerating?

Catholic Scientist's analogy is useful here:

EvC Forum: Size of the universe(In fact it's not even a full analogy, it's just making the properties of a four dimensional space/shape, the universe, more obvious by using a two-dimensional space/shape that shares some of those properties)

So the Big Bang is the North pole and every point in time after that is a circle of constant latitude. A circle of latitude being all of space at one time. As you the circles get larger the further south you go, so the spatial dimensions of the universe increase the further along in time you go.

Since the circles represent the whole universe at a given time, a point in space at a given time is like a point on a given circle.

So phrase your question in terms of this analogy:

The centre of the universe from which everything is accelerating, would be the equivalent of some special point on one of the latitude circles. However there is no special point on these circles that the circles are expanding way from, they are expanding "away" from the North Pole

Simply as a consequence of being circles on a sphere.

Similarly the universe is not expanding away from some point where the Big Bang happened. It's just that space is getting larger as you move forward in time (down the sphere).

Again, if you reverse time the circle shrinks and all points on the circle converge on the North Pole, so the North Pole "happens" at all points on the circle. Similarly the Big Bang happened everywhere.