Modern HPC architectures consist of heterogeneous multi-core, many-node systems with deep memory hierarchies. Modern applications employ ever more advanced discretisation methods to study multi-physics problems. Developing such applications that explore cutting-edge physics on cutting-edge HPC systems has become a complex task that requires significant HPC knowledge and experience. Unfortunately, this combined knowledge is currently out of reach for all but a few groups of application developers. Chemora is a framework for solving systems of Partial Differential Equations (PDEs) that targets modern HPC architectures. Chemora is based on Cactus, which sees prominent usage in the computational relativistic astrophysics community. In Chemora, PDEs are expressed either in a high-level \LaTeX-like language or in Mathematica. Discretisation stencils are defined separately from equations, and can include Finite Differences, Discontinuous Galerkin Finite Elements (DGFE), Adaptive Mesh Refinement (AMR), and multi-block systems. We use Chemora in the Einstein Toolkit to implement the Einstein Equations on CPUs and on accelerators, and study astrophysical systems such as black hole binaries, neutron stars, and core-collapse supernovae.