Analysis and Design of a New Dispersion Relation Preserving Alternate Direction Bidiagonal Compact Scheme

Abstract

In the present work bidiagonal schemes have been analyzed first by using global spectral analysis for wave propagation problem. The properties given by the numerical amplification factor, dispersion and phase error are presented for these schemes to study effects on error propagation for 1D convection equation. Analysis of bidiagonal schemes helps in designing a new alternate direction bidiagonal (ADB) spatial scheme with improved numerical dispersion and dissipation properties. Design of the ADB scheme is facilitated by the prefactorization technique given in Hixon and Turkel (J Comput Phys 158:51–70, 2000) and is obtained here by minimizing the error in computing 1D convection equation using the correct stability and error analysis given in Sengupta et al. (J Comput Phys 226:1211–1218, 2007). The ADB scheme for spatial discretization is coupled with four-stage, fourth order accurate Runge–Kutta time marching scheme and optimized for multiple objective functions by classical tools to obtain a new space-time discretization scheme with desired numerical amplification and dispersion properties up to moderate wavenumbers, preserving physical dispersion relation almost exactly. The resultant optimal DRP scheme is fourth order accurate, both in space and time. We have compared this scheme with an accurate compact scheme in reproducing exact solution of 1D convection equation. Two benchmark problems from computational acoustics are shown for further verification of the accuracy of solution obtained by the present method. To demonstrate the practical utility of the method, we have presented the solution of zero pressure gradient flow past semi-infinite flat plate from the receptivity stage to fully developed inhomogeneous turbulent state without any model and compared with published results in Sengupta et al. (Phys Rev E 85:026308, 2012).