Optimised prefactored compact schemes for linear wave propagation phenomena

A family of space- and time-optimised prefactored compact schemes are developed that minimize the computational cost for given levels of numerical error in wave propagation phenomena, with special reference to aerodynamic sound. This work extends the approach of Pirozzoli (2007) to the MacCormack type prefactored compact high-order schemes developed by Hixon (2000), in which their shorter Padè stencil from the prefactorization leads to a simpler enforcement of numerical boundary conditions. An explicit low-storage multi-step Runge-Kutta integration advances the states in time. Theoretical predictions for spatial and temporal error bounds are derived for the cost-optimised schemes and compared against benchmark schemes of current use in computational aeroacoustic applications in terms of computational cost for a given relative numerical error value. One- and two-dimensional test cases are presented to examine the effectiveness of the cost-optimised schemes for practical flow computations. An effectiveness up to about 50 \% higher than the standard schemes is verified for the linear one-dimensional advection solver, which is a popular baseline solver kernel for computational physics problems. A substantial error reduction for a given cost is also obtained in the more complex case of a two-dimensional acoustic pulse propagation, provided the optimised schemes are made to operate close to their nominal design points.