The climax can be defined in two ways. First, it can be the turning point in the action where the conflict begins to resolve itself, either positively or negatively. Second, it can be the high point of the action, the final and most exciting event in a series of events. The shattering of the conch and the death of Piggy is the event that meets both definitions in Lord of the Flies. Up until the point where Piggy dies, Ralph still believes he can reason with Jack, that there is a hope of re-establishing order and civilization on the island. He has come to Jack's base to "call an assembly" and to confront Jack about stealing Piggy's glasses. When the conch, the symbol of rules, authority, and civilization, smashes to bits, every hope Ralph has had is demolished as well. Without Piggy's assistance, Jack is significantly weakened; he has been having cloudy thinking, and Piggy has had to remind him several times of the goal of fire and rescue. Certainly this event is the high point of the action. It even occurs at the high point of the island. The tension of the near-blind Piggy clinging frantically to the rock ledge while Roger pummels him with rocks is nail-biting. When he falls to his death, it's all downhill (literally and figuratively) from there. Samneric get captured, leaving Ralph on his own to run and hide for survival, which seems unlikely with the fire raging across the island.
The resolution of the conflict occurs when the naval captain appears on the beach and talks to first Ralph and the other boys. His remark that "I should have thought that a pack of British boys . . . would have been able to put up a better show than that" drives home the resolution of the conflict. Ralph has been unable to maintain civilized society on the island; order has failed, chaos has triumphed. Golding spares us the horror of the natural end the conflict would have produced by having the boys "rescued" just in time. Yet we also know that the outside world itself is wracked with nuclear war, so the boys' rescue is ironic, and the resolution of the triumph of savagery over civilization stands.
As a study of English theater, I would argue that the climax of the novel occurs when Simon is killed. According to standard format, the climax (or conflict) is the turning point in the novel for the central character. If we view Ralph as the central character (which I think we have to), the killing of Simon represents the demise of the last little bit of leadership he had; thus, the completion in the change of power. Everything that then occurs from Simon's death until Ralph is saved falls under the category of falling action. Of course, the naval officer finding the boys and ending their war is most certainly the resolution.