The Closest Point Method (CPM) is a method for numerically solving partial differential equations (PDEs) on arbitrary surfaces, independent of the existence of a surface parametrization. The CPM uses a closest point representation of the surface, to solve the unmodified Cartesian version of a surface PDE in a 3D volume embedding, using simple and well-understood techniques. In this paper, we present the numerical solution of the wave equation and the incompressible Navier-Stokes equations on surfaces via the CPM, and we demonstrate surface appearance and shape variations in real-time using this method. To fully exploit the potential of the CPM, we present a novel GPU realization of the entire CPM pipeline. We propose a surface-embedding adaptive 3D spatial grid for efficient representation of the surface, and present a high-performance approach using CUDA for converting surfaces given by triangulations into this representation. For real-time performance, CUDA is also used for the numerical procedures of the CPM. For rendering the surface (and the PDE solution) directly from the closest point representation without the need to reconstruct a triangulated surface, we present a GPU ray-casting method that works on the adaptive 3D grid.