Study of the Error and Efficiency of Numerical Schemes for Computational Aeroacoustics

Two types of high-order numerical schemes suitable for computational aeroacoustics are examined. Truncation error, efficiency, and the consequence of disparate temporal and spatial accuracy are discussed. The Gottlieb-Turkel 2-4 predictor-corrector scheme and two Runge-Kutta schemes (4-4 and 4-6) are used to solve the one-dimensional inviscid convection of a Gaussian pulse. For schemes with lower-order time stepping, the truncation error caused by the time stepping dominates the solution for optimum time steps. Reducing the time step can effectively increase the order of accuracy to that of the spatial discretization. However, this increased accuracy is balanced by an increase in the computational cost. The uniformly fourth-order-accurate Runge-Kutta scheme proves to be superior to the second-order temporal and fourth-order spatial accurate Gottlieb-Turkel scheme in terms of truncation error and computational efficiency. Increasing the spatial accuracy of the Runge-Kutta scheme to sixth order does not improve the efficiency of the scheme. To illustrate the relevance to a representative multidimensional problem, the Gottlieb-Turkel 2-4 and Runge-Kutta 4-4 schemes are then used to solve the unsteady axisymmetric Navier-Stokes equations for a supersonic jet. For this case the Runge-Kutta scheme provides better resolution of large-scale structures and requires less computational time.