A computational model has been developed to predict and ultimately control the unsteady dynamics of aeroelastic systems. Through a general, curvilinear coordinate transformation, the model achieves exact coupling between the fluid and structure without compromising the effects of viscosity, separation, shocks, and shock-boundary layer interaction. The computational method enables a comprehensive approach to the analysis of coupled-system dynamics: capturing weak nonlinearities in an eigensystem formulation, strong nonlinearities using the full nonlinear system and dynamical systems methods, and detailed flow dynamics in time-marched simulations of the fully coupled model. In the eigensystem formulation, eigenvalues correspond to modal frequencies in the time domain and their growth rates. System stability is accurately and efficiently predicted without the need of time marching by extracting critical eigenmodes from the full, linearized system through reduced-order modeling. Validation of the eigensystem is achieved by comparing data from simulations with known, exact solutions and with time-marched solutions of the coupled fluid-structure system. Maximum relative differences between known, exact solutions and data from the model are less than one percent.