Implicit–Explicit Runge–Kutta Methods for Fast–Slow Wave Problems

Abstract

Abstract Linear multistage (Runge–Kutta) implicit–explicit (IMEX) time integration schemes for the time integration of fast-wave–slow-wave problems for which the fast wave has low amplitude and need not be accurately simulated are investigated. The authors focus on three-stage, second-order schemes and show that a scheme recently proposed by one of them (Kar) is unstable for purely oscillatory problems. The instability is reduced if the averaging inherent in the implicit part of the scheme is decentered, sacrificing second-order accuracy. Two alternative schemes are proposed with better stability properties for purely oscillatory problems. One of these utilizes a 3-cycle Lorenz scheme for the slow-wave terms and a trapezoidal scheme for the fast-wave terms. The other is a combination of two previously proposed schemes, which is stable for purely oscillatory problems for all fast-wave frequencies when the slow-wave frequency is less than a critical value. The alternative schemes are tested using a global spectral shallow-water model and a version of the NCEP operational global forecast model. The accuracy and stability of the alternative schemes are discussed, along with their computational efficiency.