RK-IMEX HEVI schemes for fully-compressible atmospheric models with advection: analyses and numerical testing
Colavolpe, Charles ; Voitus, Fabrice ; Bénard, Pierre
Année de publication
2017
<span style="color:#4b0082;"><font face="Times New Roman, serif"><font style="font-size: 12pt" size="3">Abstract</font></font></span></h2><p align="justify"><span style="color:#4b0082;">The 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, 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,...). This paper 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 analysed in two variants "UFPreF" and "UFPreB" recently proposed in the literature. The propagation of gravity waves is found to be generally well represented, but the advection makes unstable all UFPreB variants, which were on the contrary more stable without advection. The instability is analysed 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.</span></p>
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