An Advection Scheme using Paired Explicit Runge-Kutta Time Integration for Atmospheric Modeling

Abstract

In this paper， a new numerical scheme was proposed to solve the advection equation in a multi-moment nonhydrostatic dynamical core.To guarantee the shape-preserving property， the limiting operations are devised for a hybrid discretization framework adopted by the multi-moment dynamical core， consisting of the multi-moment finite-volume and the conservative finite-difference schemes for the horizontal and vertical discretizations respectively.In the horizontal direction， a nonoscillaory scheme is accomplished by adjusting the slope of the multi-moment reconstruction polynomial at the cell center with the application of a WENO （weighted essentially non-oscillatory） algorithm.The resulting multi-moment scheme can achieve the fourth-order accuracy in the convergence test.In the vertical direction， a TVD （total variation diminishing） slope limiter is applied in the finite-difference discretization to remove the non-physical oscillations around the discontinuities.To accomplish the time marching in the proposed advection model， a second-order paired explicit Runge-Kutta scheme is adopted， which is expected to be an efficient and practical method for the advection solvers in the atmospheric models with very high spatial resolutions.The explicit time marching， without the dimension splitting， is useful to avoid the divergence errors in the advection transport calculations.Two Runge-Kutta schemes， requiring different times of conducting the spatial discretization within a time step， are combined， and used for the time marching in the different directions.The finite-difference discretization is called for six times within a time step in order to increase the maximum available CFL （Courant-Friedrichs-Lewy） number in the vertical direction， while the horizontal multi-moment spatial discretization is conducted for two times as the regular second-order schemes.As a result， the difference between the maximum time steps determined by the horizontal and vertical discretizations， due to the very large aspect ratio of the computational cells in atmospheric modeling， can be diminished.The non-negativity property of the proposed advection scheme is assured by devising a new flux-correction algorithm.It improves the existing positivity-preserving algorithm through further considering the mass flowing into the computational cell in an iterative procedure during the flux-correction operations.The proposed flux-correction algorithm can approach the necessary and sufficient condition for assuring the non-negative solutions and is more accurate for the advection calculations with CFL numbers larger than one.The widely used two-dimensional benchmark tests were checked in this study and the numerical results verified the performance the proposed advection scheme， which has the practical potential to build an accurate and efficient advection equation solver for the scalable high-resolution nonhydrostatic atmospheric models.

Keywords

• advection equation
• paired explicit runge-kutta scheme
• slope limiter
• non-negative scheme
• multi-moment method