RK‐IMEX HEVI schemes for fully compressible atmospheric models with advection: analyses and numerical testing

Integration of the fully compressible non-hydrostatic equations with horizontally implicit schemes creates scalability problems for massively parallel computing architectures. An alternative is to use horizontally explicit and vertically implicit (HEVI) approaches, where the implicit problems are only along the vertical direction. Besides this, various multi-stage implicit–explicit (IMEX) methods, based on Runge–Kutta (RK) schemes, have been developed over recent years in order to achieve time discretizations free of any computational modes and possessing specified properties (accuracy-order, number of implicit iterations, amount of storage, etc.). This article compares the analytical responses of three RK-IMEX HEVI schemes (identified as attractive for atmospheric modelling), for a linear fully compressible system supporting gravity and acoustic waves as well as advection. Each scheme is analyzed in two variants, ‘UFPreF’ and ‘UFPreB’, recently proposed in the literature. The propagation of gravity waves is found to be generally well represented, but advection makes all UFPreB variants unstable; in contrast, they were more stable without advection. The instability is analyzed in a one-dimensional framework and a new class of schemes is proposed to circumvent the problem (using four Butcher tableaux, at no extra cost). A particular member of this class is examined in detail: it is shown to be accurate and stable even with advection. Some numerical testing is provided to support the analyses in a more realistic context.

[1]  Christiane Jablonowski,et al.  Operator-Split Runge-Kutta-Rosenbrock Methods for Nonhydrostatic Atmospheric Models , 2012 .

[2]  Francis X. Giraldo,et al.  A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases , 2008, J. Comput. Phys..

[3]  Nigel Wood,et al.  Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations , 2013, J. Comput. Phys..

[4]  Louis J. Wicker,et al.  Time-Splitting Methods for Elastic Models Using Forward Time Schemes , 2002 .

[5]  Nigel Wood,et al.  An inherently mass‐conserving iterative semi‐implicit semi‐Lagrangian discretization of the non‐hydrostatic vertical‐slice equations , 2010 .

[6]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[7]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[8]  Lorenzo Pareschi,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2010, 1009.2757.

[9]  Masaki Satoh,et al.  Conservative scheme for the compressible nonhydrostatic models with the horizontally explicit and vertically implicit time integration scheme , 2002 .

[10]  Dale R. Durran,et al.  Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems , 2012 .

[11]  William C. Skamarock,et al.  Efficiency and Accuracy of the Klemp-Wilhelmson Time-Splitting Technique , 1994 .

[12]  Emil M. Constantinescu,et al.  Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA) , 2013, SIAM J. Sci. Comput..

[13]  Nigel Wood,et al.  Numerical analyses of Runge–Kutta implicit–explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models , 2014 .