Review on development of a scalable high-order nonhydrostatic multi-moment constrained finite volume dynamical core

Abstract In this paper we overcome a key problem in an otherwise highly potential approach to study turbulent flows, ODTLES (One‐Dimensional Turbulence Large Eddy Simulation). From a methodological point of view, ODTLES is an approach in between Direct Numerical Simulations (DNS) and averaged/filtered approaches like RANS (Reynolds Averaged Navier‐Stokes) or LES (Large Eddy Simulations). In ODTLES, a set of 1D ODT models is embedded in a coarse grained 3D LES. On the ODT scale, the turbulent advection is modeled as a sequence of stochastic eddy events, also known as triplet maps, while the other (deterministic) terms are fully resolved in space (along the ODT‐line) and time. Schmidt et al. first (2008) introduced ODTLES and Gonzalez et al. (2011) applied the model for a variety of wall‐bounded flow problems. Although the results were notable for a first proof of concept, the numerical methods used are subject to debate. First of all, an unstable discretization for the large scale 3D ad …

A nonhydrostatic dynamical core has been developed by using the multi-moment finite volume method that ensures the rigorous numerical conservation. To represent the spherical geometry free of polar problems, the cubed-sphere grid is adopted. A fourth-order multi-moment discretization formulation is applied to the nonhydrostatic governing equations cast in local curvilinear coordinates on each patch of cubed sphere through a gnomonic projection. In vertical direction, the height-based terrain-following grid is used to represent the topography. To get around the CFL stability restriction imposed by relatively small grid spacing in the vertical direction, the dimensional-splitting time integration using the HEVI (Horizontal Explicit and Vertical Implicit) strategy is implemented by applying the IMEX Runge-Kutta scheme. The proposed dynamical core preserves the fourth-order accuracy in spherical geometry and has been verified by the widely-used benchmark tests. The results of our numerical …

Abstract The nonhydrostatic High‐Order Method Modeling Environment (HOMME‐NH) atmospheric dynamical core supports acoustic waves that propagate significantly faster than the advective wind speed, thus greatly limiting the time step size that can be used with standard explicit time integration methods. Resolving acoustic waves is unnecessary for accurate climate and weather prediction. This numerical stiffness is addressed herein by considering implicit‐explicit additive Runge‐Kutta (ARK IMEX) methods that can treat the acoustic waves in a stable manner without requiring implicit treatment of nonstiff modes. Various ARK IMEX methods are evaluated for their efficiency in producing accurate solutions, ability to take large time step sizes, and sensitivity to grid cell length ratio. Both the gravity wave test and baroclinic instability test from the 2012 Dynamical Core Model Intercomparison Project are used to recommend 5 of the 27 ARK IMEX methods tested for use in HOMME‐NH.

A discontinuous Galerkin time-domain (DGTD) method based on dynamically adaptive Cartesian mesh (ACM) is developed for a full-wave analysis of electromagnetic (EM) fields. The benefits of hierarchical Cartesian grids and adaptive mesh refinement are demonstrated for linear EM propagation problems. The developed DGTD-ACM achieves a desired accuracy by refining nonconformal meshes near material interfaces to reduce stair-casing errors without sacrificing the high efficiency afforded with uniform Cartesian meshes. More importantly, DGTD-ACM can dynamically refine the mesh to resolve the local variation of the fields during propagation of EM pulses. A local time-stepping scheme is adopted to alleviate the constraint on the time-step size due to the stability condition of the explicit time integration. It is shown by numerical examples that the proposed method can achieve a good numerical accuracy and reduce the computational time effectively for linear problems of EM propagation in dispers …

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order $α\in (0,1]$ and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in th …