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This dissertation focuses on the development and application of numerical algorithms for solving compressible multiphase fluid-material interaction problems. The first part of this dissertation is motivated by the extraordinary shock-resisting ability of elastomer coating materials (e.g., polyurea) under explosive loading conditions. Their performance, however, highly depends on their dynamic interaction with the substrate (e.g., metal) and ambient fluid (e.g., air or liquid); and the detailed interaction process is still unclear. Therefore, to certify the application of these materials, a fluid-structure coupled computational framework is needed. The first part of this dissertation developes such a framework. In particualr, the hyper-viscoelastic constitutive relation of polyurea is incorporated into a high-fidelity computational framework which couples a finite volume compressible multiphase fluid dynamics solver and a nonlinear finite element structural dynamics solver. Within this framework, the fluid-structure and liquid-gas interfaces are tracked using embedded boundary and level set methods. Then, the developed computational framework is applied to study the behavior a bilayer coating–substrate (i.e., polyurea-aluminum) system under various loading conditions. The observed two-way coupling between the structure and the bubble generated in a near-field underwater explosion motivates the next part of this dissertation. The second part of this dissertation investigates the yielding and collapse of an underwater thin-walled aluminum cylinder in near-field explosions. As the explosion intensity varies by two orders of magnitude, three different modes of collapse are discovered, including one that appears counterintuitive (i.e., one lobe extending towards the explosive charge), yet has been observed in previous laboratory experiments. Because of the transition of modes, the time it takes for the structure to reach self-contact does not decrease monotonically as the explosion intensity increases. Detailed analysis of the bubble-structure interaction suggests that, in addition to the incident shock wave, the second pressure pulse resulting from the contraction of the explosion bubble also has a significant effect on the structure's collapse. The phase difference between the structural vibration and the bubble's expansion and contraction strongly influences the structure's mode of collapse. The third part focuses on the development of efficient solution algorithms for compressible multi-material flow simulations. In these simulations, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular class of methods in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation, such as the one used in the preceding parts of the dissertation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. This dissertation accelerates the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed. The performance of these acceleration methods is assessed using four example problems that exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. For all the problems, the solution of bimaterial Riemann problems is accelerated by 37 to 87 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 81 times.