Global spectral analysis of multi-level time integration schemes: Numerical properties for error analysis

An analysis is reported here for three-time level integration methods following the global spectral analysis (GSA) described in High Accuracy Computing Methods, T.K. Sengupta, Cambridge Univ. Press, USA. The focus is on the second order Adams-Bashforth (AB2) and the extrapolation in time (EXT2) methods. Careful distinction is made for the first time step at t=0 by either Euler forward or four-stage, fourth order Runge-Kutta (RK4) time schemes. The latter is used to solve a benchmark aeroacoustic problem. Several one-dimensional wave propagation models are analyzed: pure advection and advection-diffusion equations. Various spatial discretizations are discussed, including Fourier spectral method. Attention is paid to the presence of physical and numerical modes as noted in the quadratic equation obtained from the difference equation for the model 1D convection equation. It is shown that AB2 method is less stable and accurate than EXT2 method, with respect to numerical dissipation and dispersion. This is true for the methods, in which the physical mode dominates over the numerical mode. Presented analysis provides useful guide to analyze any three-time level methods.