7 research outputs found
Adaptive mesh refinement with spectral accuracy for magnetohydrodynamics in two space dimensions
We examine the effect of accuracy of high-order spectral element methods,
with or without adaptive mesh refinement (AMR), in the context of a classical
configuration of magnetic reconnection in two space dimensions, the so-called
Orszag-Tang vortex made up of a magnetic X-point centered on a stagnation point
of the velocity. A recently developed spectral-element adaptive refinement
incompressible magnetohydrodynamic (MHD) code is applied to simulate this
problem. The MHD solver is explicit, and uses the Elsasser formulation on
high-order elements. It automatically takes advantage of the adaptive grid
mechanics that have been described elsewhere in the fluid context [Rosenberg,
Fournier, Fischer, Pouquet, J. Comp. Phys. 215, 59-80 (2006)]; the code allows
both statically refined and dynamically refined grids. Tests of the algorithm
using analytic solutions are described, and comparisons of the Orszag-Tang
solutions with pseudo-spectral computations are performed. We demonstrate for
moderate Reynolds numbers that the algorithms using both static and refined
grids reproduce the pseudo--spectral solutions quite well. We show that
low-order truncation--even with a comparable number of global degrees of
freedom--fails to correctly model some strong (sup--norm) quantities in this
problem, even though it satisfies adequately the weak (integrated) balance
diagnostics.Comment: 19 pages, 10 figures, 1 table. Submitted to New Journal of Physic
A comparison of interpolation techniques for non-conformal high-order discontinuous Galerkin methods
The capability to incorporate moving geometric features within models for
complex simulations is a common requirement in many fields. Fluid mechanics
within aeronautical applications, for example, routinely feature rotating (e.g.
turbines, wheels and fan blades) or sliding components (e.g. in compressor or
turbine cascade simulations). With an increasing trend towards the
high-fidelity modelling of these cases, in particular combined with the use of
high-order discontinuous Galerkin methods, there is therefore a requirement to
understand how different numerical treatments of the interfaces between the
static mesh and the sliding/rotating part impact on overall solution quality.
In this article, we compare two different approaches to handle this
non-conformal interface. The first is the so-called mortar approach, where flux
integrals along edges are split according to the positioning of the
non-conformal grid. The second is a less-documented point-to-point
interpolation method, where the interior and exterior quantities for flux
evaluations are interpolated from elements lying on the opposing side of the
interface. Although the mortar approach has significant advantages in terms of
its numerical properties, in that it preserves the local conservation
properties of DG methods, in the context of complex 3D meshes it poses notable
implementation difficulties which the point-to-point method handles more
readily. In this paper we examine the numerical properties of each method,
focusing not only on observing convergence orders for smooth solutions, but
also how each method performs in under-resolved simulations of linear and
nonlinear hyperbolic problems, to inform the use of these methods in implicit
large-eddy simulations.Comment: 37 pages, 15 figures, 5 tables, submitted to Computer Methods in
Applied Mechanics and Engineering, revision
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 EnergĂas 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
Simulation of the Navier-Stokes Equations in Three Dimensions with a Spectral Collocation Method
This work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation
method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables,
so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms