Abstract
Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones.The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests.
Highlights
Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency
More attention has been paid to the way in which implicit and explicit time-stepping schemes are combined to form semi-implicit models of the global atmosphere [9,11,12,23]
We will start by describing the stability analysis for Runge–Kutta Implicit and explicit (IMEX) schemes applied to wave equations
Summary
Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. Ruuth and Spiteri [ARS, 1] described how implicit and explicit Runge–Kutta (RK IMEX) schemes can be combined to create stable schemes where the combined scheme is second- or third-order accurate These were tested on advectiondiffusion equations (using the implicit part for the diffusion). Three Runge–Kutta IMEX schemes were tested by Ullrich and Jablonowski [23] for the HEVI solution of the equations governing atmospheric motion They tested the ARS(2,3,2) scheme of Ascher et al [1] and suggested the less computationally expensive but nearly as accurate Strang carryover scheme. This paper considers the numerical solution of the equations governing compressible atmospheric waves It investigates RK IMEX schemes applied both to a semi-implicit approach (treating acoustic and gravity waves implicitly and advection explicitly) and to a HEVI approach (Horizontally Explicit, Vertically Implicit) (Section 2.2). The formulation of the Helmholtz equations by combining the momentum, temperature and continuity equations is described in Appendix A and the spatial discretisation is described in Appendix B
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