John Barnes' Directive 51 (ACE, 2010) is a futuristic apocalyptic novel that explores what might happen to the national government (and the rest of the world for that matter) when a major catastrophe strikes, and the president is no longer capable of continuing in his role as commander in chief.
The titular directive refers to one that actually exists today, providing "that in a case of unprecedented disaster a specific Federal official will become temporary dictator, with nearly unlimited power and a mission to restore Constitutional government as quickly as possible."
Barnes explores this in Directive 51, but does not allow the politics to bog down the rest of the book. The novel is filled with acts of terror, dramatic tension, survivalism and even romance. The characters, even those secondary to the plot, experience real change throughout the course of the novel.
Called "the thinking person's techno-thriller," by S.M. Stirling, I'd recommend this to any reader of science fiction.