Papers by Francis Giraldo
: A new dynamical core for numerical weather prediction (NWP) based on the spectral element Euler... more : A new dynamical core for numerical weather prediction (NWP) based on the spectral element Eulerian-Lagrangian (SEEL) method is presented. This paper represents a departure from previously published work on solving the atmospheric equations in that the horizontal operators are all written, discretized, and solved in 3D Cartesian space. The advantages of this new methodology are: the pole singularity which plagues all gridpoint methods disappears, the horizontal operators can be approximated by local high-order elements, the Eulerian-Lagrangian formulation permits extremely large time-steps, and the fully-implicit Eulerian-Lagrangian formulation only requires the inversion of a 2D Helmholtz operator. In order to validate the SEELAM model, results for four test cases are shown. These are: the Rossby-Haurwitz waves number 1 and 4, and the Jablonowski-Williamson balanced initial state and baroclinic instability tests. Comparisons with four well-established operational models show that ...
IMEX HDG-DG: A coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for shallow water systems
Journal of Computational Physics
Archives of Computational Methods in Engineering
The continuous partial differential equations governing a given physical phenomenon, such as the ... more The continuous partial differential equations governing a given physical phenomenon, such as the Navier-Stokes equations describing the fluid motion, must be numerically discretized in space and time in order to obtain a solution otherwise not readily available in closed (i.e., analytic) form. While the overall numerical discretization plays an essential role in the algorithmic efficiency and physically-faithful representation of the solution, the time-integration strategy commonly is one of the main drivers in terms of cost-to-solution (e.g., time-or energy-to-solution), accuracy and numerical stability, thus constituting one of the key building blocks of the computational model. This is especially true in time-critical applications, including numerical weather prediction (NWP), climate simulations and engineering. This review provides a comprehensive overview of the existing and emerging time-integration (also referred to as time-stepping) practices used in the operational global NWP and climate industry, where global refers to weather and climate simulations performed on the entire globe. While there are many flavors of time-integration strategies, in this review we focus on the most widely adopted in NWP and climate centers and we emphasize the reasons why such numerical solutions were adopted. This allows us to make some considerations on future trends in the field such as the need to balance accuracy in time with substantially enhanced time-to-solution and associated implications on energy consumption and running costs. In addition, the potential for the co-design of time-stepping algorithms and underlying high performance computing hardware, a keystone to accelerate the computational performance of future NWP and climate services, is also discussed in the context of the demanding operational requirements of the weather and climate industry.
Journal of Computational Physics
We present a novel high-order discontinuous Galerkin discretization for the spherical shallow wat... more We present a novel high-order discontinuous Galerkin discretization for the spherical shallow water equations, able to handle wetting/drying and non-conforming, curved meshes in a well-balanced manner. This requires a well-balanced discretization, that cannot rely on exact quadrature, due to the curved mesh. Using the strong form of the discontinuous Galerkin discretization, we achieve a splitting of the well-balanced condition into individual problems for the flux and volume terms, which has significant advantages: It allows for the construction of non-conforming, well-balanced flux discretizations, i.e. we can perform nonconforming mesh refinement while preserving the well-balanced property of the scheme. More importantly, this approach enables the development of a new method for handling wet/dry transitions. In contrast to other wetting/drying methods, it is well-balanced and able to handle wetting/drying robustly at any polynomial order, without the introduction of physical model assumptions such as viscosity, artificial porosity or cancellation of gravity. We perform a series of one-dimensional tests and analyze the properties of our scheme. In order to validate our method for the simulation of large-scale tsunami events on the rotating sphere, we perform numerical simulations of the 2011 Tohoku tsunami and compare our results to real-world buoy data. The method is able to predict arrival times and wave amplitudes accurately even over long distances. This indicates that our method accurately captures all physical phenomena relevant to the long-term evolution of tsunami waves.
Quarterly Journal of the Royal Meteorological Society
We test the behaviour of a unified continuous/discontinuous Galerkin (CG/DG) shallowwater model i... more We test the behaviour of a unified continuous/discontinuous Galerkin (CG/DG) shallowwater model in spherical geometry with curved elements on three different grids of ubiquitous use in atmospheric modelling: (i) the cubed-sphere, (ii) the reduced latitude-longitude, and (iii) the icosahedral grid. Both conforming and non-conforming grids are adopted including static and dynamically adaptive grids for a total of twelve mesh configurations. The behaviour of CG and DG on the different grids are compared for a nonlinear midlatitude perturbed jet and for a linear case that admits an analytic solution. Because the inviscid solution on certain grids shows a high sensitivity to the resolution, the viscous counterpart of the governing equations is also solved and the results compared. The logically unstructured element-based CG/DG model described in this article is flexible with respect to arbitrary grids. However, we were unable to define a best grid configuration that could possibly minimize the error regardless of the characteristic geometry of the flow. This is especially true if the governing equations are not regularized by the addition of a sufficiently large, fully artificial, diffusion mechanism, as will be shown. The main novelty of this study lies in the unified implementation of two element-based Galerkin methods that share the same numerical machinery and do not rely on any specific grid configuration to solve the shallow-water equation on the sphere.
Nodal High-Order Discontinuos Galerkin Methods for the Spherical Shallow Water Equations
Advanced Numerical Methods for Numerical Weather Prediction (NWP)
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Explicit-in-time CG a continuous Galerkin discretization of the compressible Euler mini-app with ... more Explicit-in-time CG a continuous Galerkin discretization of the compressible Euler mini-app with explicit time integration; Explicit-in-time DG a discontinuous Galerkin discretization of the compressible Euler mini-app with explicit time integration; Vertically Semi-Implicit CG a continuous Galerkin discretization of the compressible Euler mini-app with vertically implicit semi-implicit time integration; Vertically Semi-Implicit DG a discontinuous Galerkin discretization of the compressible Euler mini-app with vertically implicit semi-implicit time integration; Once the performance of a mini-app is accepted it will be considered for adoption into NUMA. Extending the kernels in NUMA is being lead by Giraldo and his postdoctoral researcher Abdi. We will also make these mini-apps available to the community to be imported into other codes if desired. Wilcox is working closely with Warburton and his team to lead the effort to develop the mini-apps including hand rolled computational kernels optimized for GPU accelerators. These kernels are "hand-written" in OCCA, a library Warburton's group is developing that allows a single kernel to be compiled using many different threading frameworks, such as CUDA, OpenCL, OpenMP, and Pthreads. We are initially developing handwritten kernels to provide a performance target for the Loo.py generated kernels. Parallel communication between computational nodes will use the MPI standard to enable the mini-apps to run on large scale clusters. Using these community standards for parallel programing will allow our mini-apps to be portable to many platforms, however the performance may not be portable across devices. For performance portability, we, lead by Klöckner, are using Loo.py to generate OCCA kernels which can be automatically tuned for current many-core devices along with future ones.
Geoscientific Model Development Discussions
In this paper, we present a dynamical core for the atmospheric primitive hydrostatic equations us... more In this paper, we present a dynamical core for the atmospheric primitive hydrostatic equations using a unified formulation of spectral element (SE) and discontinuous Galerkin (DG) methods in the horizontal direction with a finite difference (FD) method in the radial direction. The CG and DG horizontal discretization employs high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points, which define the common machinery. The atmospheric primitive hydrostatic equations are solved on the cubed-sphere grid using the flux form governing equations in a three-dimensional (3-D) Cartesian space. By using Cartesian space, we can avoid the pole singularity problem due to spherical coordinates and this also allows us to use any quadrilateral-based grid naturally. In order to consider an easy way for coupling the dynamics with existing physics packages, we use a FD in the radial direction. The models are verified by conducting conve...
Geoscientific Model Development Discussions
The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplifie... more The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density curren...
Journal of the Atmospheric Sciences
The fundamental pathways for tropical cyclone (TC) intensification are explored by considering ax... more The fundamental pathways for tropical cyclone (TC) intensification are explored by considering axisymmetric and asymmetric impulsive thermal perturbations to balanced, TC-like vortices using the dynamic cores of three different nonlinear numerical models. Attempts at reproducing the results of previous work, which used the community WRF Model, revealed a discrepancy with the impacts of purely asymmetric thermal forcing. The current study finds that thermal asymmetries can have an important, largely positive role on the vortex intensification, whereas other studies find that asymmetric impacts are negligible. Analysis of the spectral energetics of each numerical model indicates that the vortex response to asymmetric thermal perturbations is significantly damped in WRF relative to the other models. Spectral kinetic energy budgets show that this anomalous damping is primarily due to the increased removal of kinetic energy from the vertical divergence of the vertical pressure flux, whic...
The long-term goal of this research is to construct global and mesoscale nonhydrostatic numerical... more The long-term goal of this research is to construct global and mesoscale nonhydrostatic numerical weather prediction (NWP) models for the U.S. Navy using new numerical methods specifically designed for modern computer architectures. To take full advantage of distributed-memory computers, the global domains of these new models are partitioned into local sub-domains, or elements, that can then be solved independently on multiple processors. The numerical methods used on these subdomains are local, high-order accurate, fully conservative, and highly efficient. Using these ideas we are developing global and mesoscale nonhydrostatic atmospheric models that will improve upon the operational models currently used by all U.S. agencies including the U.S. Navy.
Partial Observability and its Consistency for PDEs
Limited area doomain: 160 x 120 x 24 km Resolution: dx = 1000 m dz_min = 150 m (at z=0) dz_max = ... more Limited area doomain: 160 x 120 x 24 km Resolution: dx = 1000 m dz_min = 150 m (at z=0) dz_max = 550 m Gray iso-surface of qc = 2 g/kg shading of cloud water, qc-Color shading perturbation potential temperature,-Uniform resolution: 25 m
Agu Fall Meeting Abstracts, Dec 1, 2003
A new dynamical core for numerical weather prediction (NWP) based on the spectral element method ... more A new dynamical core for numerical weather prediction (NWP) based on the spectral element method is presented. This paper represents a departure from previously published work on solving the atmospheric primitive equations in that the horizontal operators are all written, discretized, and solved in 3D Cartesian space. The advantages of using Cartesian space are that the pole singularity that plagues the equations in spherical coordinates disappears; any grid can be used, including latitude-longitude, icosahedral, hexahedral, and adaptive unstructured grids; and the conversion to a semi-Lagrangian formulation is easily achieved. The main advantage of using the spectral element method is that the horizontal operators can be approximated by local high-order elements while scaling efficiently on distributed-memory computers. In order to validate the 3D global atmospheric spectral element model, results are presented for seven test cases: three barotropic tests that confirm the exponential accuracy of the horizontal operators and four baroclinic test cases that validate the full 3D primitive hydrostatic equations. These four baroclinic test cases are the Rossby-Haurwitz wavenumber 4, the Held-Suarez test, and the Jablonowski-Williamson balanced initial state and baroclinic instability tests. Comparisons with four operational NWP and climate models demonstrate that the spectral element model is at least as accurate as spectral transform models while scaling linearly on distributed-memory computers.
The solution of the Euler equations by the spectral element method (SEM) is subject to oscillator... more The solution of the Euler equations by the spectral element method (SEM) is subject to oscillatory behavior if the high-frequency modes are not damped in some way. In this analysis, we extend to high order spectral elements and to low-Mach number flows the recent work by Nazarov and Hoffman [20], where an LES-like physical diffusion acts both as a localized and controlled numerical stabilization for finite elements and as a turbulence model for compressible flows. In the framework of high-order SEM for the solution of the low-Mach number flows, this approach is a possible physics-based alternative to the variational multiscale stabilization (VMS) method that the authors successfully applied to the SEM solution of the advection diffusion equation [17] in the context of atmospheric flows. Like for VMS, stabilization is obtained by means of a residual-based term that is added to the inviscid Euler equations. Unlike VMS, however, this extra term is based on purely physical-rather than numerical-assumptions, in that it is related to the viscous component of the stress tensor of the Navier-Stokes equations. The method is tested with pseudo and fully 3D simulations of idealized nonhydrostatic atmospheric flows and is verified against data from the literature. This work represents a step toward the implementation of a stabilized, high order, spectral element LES model within the Nonhydrostatic Unified Model of the Atmosphere (NUMA) developed by the authors.
LES density current simulation: Animated GIF
Jet on spherical adaptive grids Video by Andreas Mueller (Naval Postgraduate School)
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Papers by Francis Giraldo