Donald carded a 3-under 67. Both he and Woods stand at 4-under-par 136 through 36 holes.
Vijay Singh and Henrik Stenson, who shared the opening round lead with Woods, both shot 1-over 71 on Friday and are tied for third with Thomas Bjorn (67), Paul McGinley (66) and Chris DiMarco (70) at 3-under 137.
PGA Championship winner Phil Mickelson bogeyed two of his last five holes for a 2-over 72 and is tied for 20th with four other players at 1-over 141.
Defending champion Stewart Cink is tied for 32nd at 4-over 144.