Abstract: Scalability of codes at extreme scales is a critical issue of current relevance. At extreme levels of parallelisms, communication between processing elements(PEs) could represent a substantial running time, resulting in substantial waste in computing cycles. We investigate a novel approach based on common finite-difference schemes for partial differential equations(PDEs) in which computations are done locally with values already available at each PE, without waiting for updated data from neighboring PEs. No synchronization among cores is enforced and computations proceed regardless of messages status. This drastically reduces processor idle times, improving computation rates and scalability. We show that accuracy of common numerical schemes is reduced when asynchronicity is present. We derive new schemes that can maintain the accuracy under asynchronous conditions. These new schemes are implemented and tested. Performance is evaluated through statistics of different measures of asynchronicity. A new implementation of RDMA communications is shown to provide significant advantages .