Characters in real-time computer games need to move smoothly and thus need to search in real time. In this paper, we describe a simple but powerful way of speeding up repeated A* searches with the same goal states, namely by updating the heuristics between A* searches. We then use this technique to develop a novel real-time heuristic search method, called Real-Time Adaptive A*, which is able to choose its local search spaces in a fine-grained way. It updates the values of all states in its local search spaces and can do so very quickly. Our experimental results for characters in real-time computer games that need to move to given goal coordinates in unknown terrain demonstrate that this property allows Real-Time Adaptive A* to follow trajectories of smaller cost for given time limits per search episode than a recently proposed real-time heuristic search method that is more difficult to implement.