A numerical method is developed for application to unsteady fluid dynamics problems, in particular to the mechanics following a sudden release of high energy. Solution of the initial compressible flow phase provides input to a power-series method for the incompressible fluid motions. The system is split into spatial and time domains leading to the convergent computation of a sequence of elliptic equations. Two sample problems are solved, the first involving an underwater explosion and the second the response of a nuclear reactor containment shell structure to a hypothetical core accident. The solutions are correlated with experimental data.