Semi-implicit formulations of the Navier-Stokes equations: application to nonhydrostatic atmospheric modeling

We present semi-implicit (implicit-explicit) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit Runge-Kutta method for all the tests studied, especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including inertia-gravity waves, density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and nonreflecting boundary conditions (NRBCs). It is shown that the NRBCs exert a strong impact on the accuracy and efficiency of the models.Office of Naval ResearchJunta de AndalucíaGerman Research Foundatio

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

Numerical weather prediction (NWP) is in a period of transition. As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where it will be necessary to use a single model for both applications. The new dynamical cores at the heart of these unified models are designed to scale efficiently on clusters with hundreds of thousands or even millions of CPU cores and GPUs. Operational and research NWP codes currently use a wide range of numerical methods: finite differences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin, which constitute element-based Galerkin (EBG) methods.Due to their important role in this transition, will EBGs be the dominant power behind NWP in the next 10 years, or will they just be one of many methods to choose from? One decade after the review of numerical methods for atmospheric modeling by Steppeler et al. (Meteorol Atmos Phys 82:287–301, 2003), this review discusses EBG methods as a viable numerical approach for the next-generation NWP models. One well-known weakness of EBG methods is the generation of unphysical oscillations in advection-dominated flows; special attention is hence devoted to dissipation-based stabilization methods. Since EBGs are geometrically flexible and allow both conforming and non-conforming meshes, as well as grid adaptivity, this review is concluded with a short overview of how mesh generation and dynamic mesh refinement are becoming as important for atmospheric modeling as they have been for engineering applications for many years.The authors would like to thank Prof. Eugenio Oñate (U. Politècnica de Catalunya) for his invitation to submit this review article. They are also thankful to Prof. Dale Durran (U. Washington), Dr. Tommaso Benacchio (Met Office), and Dr. Matias Avila (BSC-CNS) for their comments and corrections, as well as insightful discussion with Sam Watson, Consulting Software Engineer (Exa Corp.) Most of the contribution to this article by the first author stems from his Ph.D. thesis carried out at the Barcelona Supercomputing Center (BSCCNS) and Universitat Politècnica de Catalunya, Spain, supported by a BSC-CNS student grant, by Iberdrola Energías Renovables, and by grant N62909-09-1-4083 of the Office of Naval Research Global. At NPS, SM, AM, MK, and FXG were supported by the Office of Naval Research through program element PE-0602435N, the Air Force Office of Scientific Research through the Computational Mathematics program, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. The scalability studies of the atmospheric model NUMA that are presented in this paper used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. SM, MK, and AM are grateful to the National Research Council of the National Academies.Peer ReviewedPostprint (author's final draft

An IMEX-DG solver for atmospheric dynamics simulations with adaptive mesh refinement

We present an accurate and efficient solver for atmospheric dynamics simulations that allows for non-conforming mesh refinement. The model equations are the conservative Euler equations for compressible flows. The numerical method is based on an $h-$adaptive Discontinuous Galerkin spatial discretization and on a second order Additive Runge Kutta IMEX method for time discretization, especially designed for low Mach regimes. The solver is implemented in the framework of the $deal.II$ library, whose mesh refinement capabilities are employed to enhance efficiency. A number of numerical experiments based on classical benchmarks for atmosphere dynamics demonstrate the properties and advantages of the proposed method

Addressing the challenges of implementation of high-order finite volume schemes for atmospheric dynamics of unstructured meshes

The solution of the non-hydrostatic compressible Euler equations using Weighted Essentially Non-Oscillatory (WENO) schemes in two and three-dimensional unstructured meshes, is presented. Their key characteristics are their simplicity; accuracy; robustness; non-oscillatory properties; versatility in handling any type of grid topology; computational and parallel efficiency. Their defining characteristic is a non-linear combination of a series of high-order reconstruction polynomials arising from a series of reconstruction stencils. In the present study an explicit TVD Runge-Kutta 3rd -order method is employed due to its lower computational resources requirement compared to implicit type time advancement methods. The WENO schemes (up to 5th -order) are applied to the two dimensional and three dimensional test cases: a 2D rising

Simulation of all-scale atmospheric dynamics on unstructured meshes

The advance of massively parallel computing in the nineteen nineties and beyond encouraged finer grid intervals in numerical weather-prediction models. This has improved resolution of weather systems and enhanced the accuracy of forecasts, while setting the trend for development of unified all-scale atmospheric models. This paper first outlines the historical background to a wide range of numerical methods advanced in the process. Next, the trend is illustrated with a technical review of a versatile nonoscillatory forward-in-time finite-volume (NFTFV) approach, proven effective in simulations of atmospheric flows from small-scale dynamics to global circulations and climate. The outlined approach exploits the synergy of two specific ingredients: the MPDATA methods for the simulation of fluid flows based on the sign-preserving properties of upstream differencing; and the flexible finite-volume median-dual unstructured-mesh discretisation of the spatial differential operators comprising PDEs of atmospheric dynamics. The paper consolidates the concepts leading to a family of generalised nonhydrostatic NFTFV flow solvers that include soundproof PDEs of incompressible Boussinesq, anelastic and pseudo-incompressible systems, common in large-eddy simulation of small- and meso-scale dynamics, as well as all-scale compressible Euler equations. Such a framework naturally extends predictive skills of large-eddy simulation to the global atmosphere, providing a bottom-up alternative to the reverse approach pursued in the weather-prediction models. Theoretical considerations are substantiated by calculations attesting to the versatility and efficacy of the NFTFV approach. Some prospective developments are also discussed

IMplicit-EXplicit Formulations for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models

This work presents IMplicit-EXplicit (IMEX) formulations for discontinuous Galerkin (DG) discretizations of the compressible Euler equations governing non-hydrostatic atmospheric flows. In particular, we show two different IMEX formulations that not only treat the stiffness due to the governing dynamics but also the domain discretization. We present these formulations for two different equation sets typically employed in atmospheric modeling. For both equation sets, efficient Schur complements are derived and the challenges and remedies for deriving them are discussed. The performance of these IMEX formulations of different orders are investigated on both 2D (box) and 3D (sphere) test problems and shown to achieve their theoretical rates of convergence and their efficiency with respect to both mesoscale and global applications are presented

Two-dimensional evaluation of atham-fluidity, a nonhydrostatic atmospheric model using mixed continuous/discontinuous finite elements and anisotropic grid optimization

AbstractThis paper presents the first attempt to apply the compressible nonhydrostatic Active Tracer High-Resolution Atmospheric Model–Fluidity (ATHAM-Fluidity) solver to a series of idealized atmospheric test cases. ATHAM-Fluidity uses a hybrid finite-element discretization where pressure is solved on a continuous second-order grid while momentum and scalars are computed on a first-order discontinuous grid (also known as ). ATHAM-Fluidity operates on two- and three-dimensional unstructured meshes, using triangular or tetrahedral elements, respectively, with the possibility to employ an anisotropic mesh optimization algorithm for automatic grid refinement and coarsening during run time. The solver is evaluated using two-dimensional-only dry idealized test cases covering a wide range of atmospheric applications. The first three cases, representative of atmospheric convection, reveal the ability of ATHAM-Fluidity to accurately simulate the evolution of large-scale flow features in neutral atmospheres at rest. Grid convergence without adaptivity as well as the performances of the Hermite–Weighted Essentially Nonoscillatory (Hermite-WENO) slope limiter are discussed. These cases are also used to test the grid optimization algorithm implemented in ATHAM-Fluidity. Adaptivity can result in up to a sixfold decrease in computational time and a fivefold decrease in total element number for the same finest resolution. However, substantial discrepancies are found between the uniform and adapted grid results, thus suggesting the necessity to improve the reliability of the approach. In the last three cases, corresponding to atmospheric gravity waves with and without orography, the model ability to capture the amplitude and propagation of weak stationary waves is demonstrated. This work constitutes the first step toward the development of a new comprehensive limited area atmospheric model.This research has received funding from the European Union Seventh Framework Program (FP7/2007-2013) under Grant agreement 603663 for the research project PEARL (Preparing for Extreme And Rare events in coastaL regions). The EPSRC multiphase program grant MEMPHIS is also acknowledged.This is the author accepted manuscript. The final version is available from the American Meteorological Society via http://dx.doi.org/10.1175/MWR-D-15-0398.

Comparison of the ENATE approach and discontinuous Galerkin spectral element method in 1D nonlinear transport equations

In this paper a comparison of the performance of two ways of discretizing the nonlinear convection-diffusion equation in a one-dimensional (1D) domain is performed. The two approaches can be considered within the class of high-order methods. The first one is the discontinuous Galerkin method, which has been profusely used to solve general transport equations, either coupled as the Navier-Stokes equations, or on their own. On the other hand, the ENATE procedure (Enhanced Numerical Approximation of a Transport Equation), uses the exact solution to obtain an exact algebraic equation with integral coefficients that link nodal values with a three-point stencil. This paper is the first of thorough assessments of ENATE by comparing it with well-established high-order methods. Several test cases of the steady Burgers' equation with and without source have been chosen for comparison