Convection experiments with the exponential time integration scheme

Abstract

We present the construction and preliminary tests of the exponential time integration schemes for the Euler equations. The equations are written in terms of potential temperature and Exner function. The time stepping is accomplished with the EPI2 and EPI3 schemes which have been used in the past for very accurate, long time step integration of the shallow water equations on the sphere. The stability of exponential integrators of this class is ensured by approximation of the oscillatory term using an absolutely stable exponential formula. Consequently, the schemes are not subject to the CFL condition imposed by the linear oscillatory part of the system and they also eliminate the phase errors associated with all the known implicit time stepping algorithms. All calculations are performed using the phipm algorithm based on the Krylov space methods. It is shown that the spectral radius of the Jacobian and consequently the Krylov space dimension are very sensitive to the presence of spatial noise. Efficiency and accuracy increase after the Shapiro filter is applied every two time steps. The method is investigated using several standardized and well accepted benchmark tests, including convective bubbles and a cold density current proposed by Straka and collaborators. Numerical experiments show that a stable integration is obtained for the wave Courant number of the order of 60 to 250. We present a detailed discussion of the development of a cold density current capturing the spiraling Prandtl vortex and the subsequent appearance of the Kelvin–Helmholtz billows. Our results from the exponential method compare very well with those obtained from the accurate schemes reported in the recent literature. The algorithm accurately handles both advection as well as acoustic and gravity waves. These results warrant the further development of the scheme for modeling atmospheric processes on a wide range of scales.