A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

Giraldo, Francis X.; Restelli, Marco

doi:10.1016/j.jcp.2007.12.009

Cited by 225 publications

(399 citation statements)

References 55 publications

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“…Therefore, it is of paramount importance that state-of-the-art NWP models account for non-hydrostatic effects with increased resolution. This work is conducted using the Non-hydrostatic Unified Model of the Atmosphere (NUMA) [14] which we believe is well placed to meet this goal. NUMA has the following features: a) non-hydrostatic, b) high-order element based Galerkin (EBG) methods for spatial discretization, c) unified regional and global NWP, d) static/dynamic Adaptive Mesh Refinement (AMR), e) high-order explicit, implicit-explicit (IMEX) and fully implicit temporal discretization, f) scalable to millions of CPUs [29] and thousands of GPUs [39].…”

Section: Introductionmentioning

confidence: 99%

“…Within the realm of atmospheric modeling, the first use of high order CG appeared in [17,12] and that of high order DG methods in [14,30]. CG has been the most popular choice for solving Partial Differential Equations (PDEs) in solid mechanics, however, DG is becoming more popular in the computational fluid dynamics field due to several desirable properties it possesses [31].…”

Section: Introductionmentioning

confidence: 99%

“…The implementations are validated with several examples that are representative of typical atmospheric dynamics. Cloud resolving capabilities are tested with a rising thermal bubble problem [14], mesoscale modeling capabilities with a density current problem [37,13] and a global scale capability is tested with an acoustic wave propagation problem run on the entire planet [38].…”

Section: Introductionmentioning

confidence: 99%

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Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Abdi

Giraldo

2016

Journal of Computational Physics

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A unified approach for the numerical solution of the 3D hyperbolic Euler equations using high order methods, namely continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, is presented. First, we examine how classical CG that uses a global storage scheme can be constructed within the DG framework using constraint imposition techniques commonly used in the finite element literature. Then, we implement and test a simplified version in the Non-hydrostatic Unified Model of the Atmosphere (NUMA) for the case of explicit time integration and a diagonal mass matrix. Constructing CG within the DG framework allows CG to benefit from the desirable properties of DG such as, easier hp-refinement, better stability etc. Moreover, this representation allows for regional mixing of CG and DG depending on the flow regime in an area. The different flavors of CG and DG in the unified implementation are then tested for accuracy and performance using a suite of benchmark problems representative of cloud-resolving scale, meso-scale and global-scale atmospheric dynamics. The value of our unified approach is that we are able to show how to carry both CG and DG methods within the same code and also offer a simple recipe for modifying an existing CG code to DG and vice versa.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Abdi

Giraldo

2016

Journal of Computational Physics

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“…The system of Eq. (1) is just one of several possibilities to write the equations governing the atmosphere (for alternatives see, for instance, [41,42]). In addition, here we do not formally define the physical parametrization terms P and Q, as they are not relevant for the purpose of this review.…”

Section: Equations Modeling the Atmospherementioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

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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.

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“…We understand that the naming convention of strong versus weak is somewhat unusual, but we point out that we use this convention here in order to remain consistent with the terminology in our previous work (e.g. [9,25,30,31]). It should be understood that the weak and strong forms mentioned here both represent weak formulations of the continuous problem (perhaps weak form and strong weak form are more apt).…”

Section: Semi-discrete Equationsmentioning

confidence: 99%

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

Giraldo

Restelli

2010

Numerical Methods in Fluids

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SUMMARYWe extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders one to six. The reason for changing the time-integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high-order DG methods. Changing the timeintegration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element-type area integrals, but also the finite volume-type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high-order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high-order (HO) time-integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low-order time discretizations. Published in

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A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

Cited by 225 publications

References 55 publications

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

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