Solution of linearized rotating shallow water equations by compact schemes with different grid-staggering strategies

Abstract

High accuracy solution of PDEs requires proper error analysis. Previous analysis for a non-dispersive system [T.K. Sengupta, A. Dipankar, P. Sagaut, Error dynamics: beyond von Neumann analysis, J. Comput. Phys. 226 (2007) 1211–1218] identified sources of error correctly. Here, the aim is to extend the spectral analysis for the model linearized rotating shallow water equations (LRSWE), as an example of dispersive system. We perform the analysis when high accuracy compact schemes are used to solve the LRSWE relevant to geophysical fluid dynamics, using different grid arrangements proposed in Mesinger and Arakawa [F. Mesinger, A. Arakawa, Numerical Methods Used in Atmospheric Models, GARP Publ. Ser. No. 17, vol. 1, WMO, Geneva, 1976, pp. 43–64] and Randall [D.A. Randall, Geostrophic adjustment and the finite-difference shallow-water equations, Mon. Wea. Rev. 122 (1994) 1371–1377]. Compact schemes are used for fluid dynamical problem, as these afford near-spectral accuracy in solving non-periodic problems. However, higher accuracy methods also suffer from errors, those are often filtered by low order methods. For example, dispersion error is present in all numerical methods and extreme form of it leads to q-waves, which appear at higher wavenumbers for compact schemes as compared to lower order method. We also evaluate a compact scheme specifically designed for use with staggered grids. Here, two and four time-level temporal discretization methods have been compared for solving LRSWE by considering classical fourth-order, four-stage Runge–Kutta ( RK 4), two time-level forward–backward (FB) and four time-level generalized FB temporal integration schemes.