As the discipline of scientific computing grows, so too does the "skills gap" between the increasingly complex scientific applications and the efficient algorithms required. Increasing demand for computational power on the march towards exascale requires innovative approaches. Closing the skills gap avoids the many pitfalls that lead to poor utilisation of resources and wasted investment. This thesis tackles two challenges: asynchronous algorithms for parallel computing and fault tolerance. First I present a novel asynchronous task invocation methodology for Discontinuous Galerkin codes called enclave tasking. The approach modifies the parallel ordering of tasks that allows for efficient scaling on dynamic meshes up to 756 cores. It ensures high levels of concurrency and intermixes tasks of different computational properties. Critical tasks along domain boundaries are prioritised for an overlap of computation and communication. The second contribution is the teaMPI library, forming teams of MPI processes exchanging consistency data through an asynchronous "heartbeat". In contrast to previous approaches, teaMPI operates fully asynchronously with reduced overhead. It is also capable of detecting individually slow or failing ranks and inconsistent data among replicas. Finally I provide an outlook into how asynchronous teams using enclave tasking can be combined into an advanced team-based diffusive load balancing scheme. Both concepts are integrated into and contribute towards the ExaHyPE project, a next generation code that solves hyperbolic equation systems on dynamically adaptive cartesian grids
