Physical modeling synthesis represents an exciting frontier in music production and digital audio. Unlike traditional multisampling-based approaches, which deliver realism and variety at the cost of extensive storage, physics-based modelling offers the potential to recreate the complex sounds of acoustic instruments through computational simulation rather than recording and replaying. This opens up the possibility to generate highly dynamic, realistic and responsive sounds by simulating the physical mechanisms responsible for generating sounds from their respective instruments. Among these, drums present a particularly interesting challenge due to the complex interactions between vibrating membranes and resonant bodies, and as a drummer myself, their simulation will be the focus of this project. However, the computational demands of accurate physical modeling have generally limited its application, especially for real-time interactive use. The simulation of a vibrating membrane alone requires solving complex differential equations across thousands of spatial points and time steps, and this computational barrier has historically meant that high-fidelity physics-based drum synthesis has remained primarily in the domain of academic research or offline rendering. The main goal of this project has been to tackle this computational challenge by utilising two complementary approaches; leveraging the massive parallelism offered by GPU-based computation, and developing approximations of the physical mechanisms that preserve accuracy to an acceptable level whilst further reducing computational complexity.