We present a flexible and scalable method to compute global solutions of high-dimensional non-smooth dynamic models. Within a time-iteration setup, we interpolate policy functions using an adaptive sparse grid algorithm with piecewise multi-linear (hierarchical) basis functions. As the dimensionality increases, sparse grids grow considerably slower than standard tensor product grids. In addition, the grid scheme we use is automatically refined locally and can thus capture steep gradients or even non-differentiabilities. To further increase the maximal problem size we can handle, our implementation is fully hybrid parallel, i.e. using a combination of MPI and OpenMP. This parallelization enables us to efficiently use modern high-performance computing architectures. Our time iteration algorithm scales up nicely to more than one thousand parallel processes. To demonstrate the performance of our method, we apply it to high-dimensional international real business cycle models with capital adjustment costs and irreversible investment.