scholarly journals Islet: Interpolation semi-Lagrangian element-based transport

2021 ◽  
Author(s):  
Andrew M. Bradley ◽  
Peter A. Bosler ◽  
Oksana Guba
Keyword(s):  
Test Problem ◽  
Stability Limit ◽  
Spectral Element ◽  
Tracer Transport ◽  
Time Step ◽  
Trace Species ◽  
Diagnostic Values ◽  
Passive Tracers ◽  
Performance Results ◽  
Eulerian Methods

Abstract. Advection of trace species (tracers), also called tracer transport, in models of the atmosphere and other physical domains is an important and potentially computationally expensive part of a model's dynamical core (dycore). Semi-Lagrangian (SL) advection methods are efficient because they permit a time step much larger than the advective stability limit for explicit Eulerian methods. Thus, to reduce the computational expense of tracer transport, dycores often use SL methods to advect passive tracers. The class of interpolation semi-Lagrangian (ISL) methods contains potentially extremely efficient SL methods. We describe a set of ISL bases for element-based transport, such as for use with atmosphere models discretized using the spectral element (SE) method. An ISL method that uses the natural polynomial interpolant on Gauss-Legendre-Lobatto (GLL) SE nodes of degree at least three is unstable on the test problem of periodic translational flow on a uniform element grid. We derive new alternative bases of up to order of accuracy nine that are stable on this test problem; we call these the Islet bases. Then we describe an atmosphere tracer transport method, the Islet method, that uses three grids that share an element grid: a dynamics grid supporting, for example, the GLL basis of degree three; a physics grid with a configurable number of finite-volume subcells per element; and a tracer grid supporting use of our Islet bases, with particular basis again configurable. This method provides extremely accurate tracer transport and excellent diagnostic values in a number of validation problems. We conclude with performance results that use up to 27,600 NVIDIA V100 GPUs on the Summit supercomputer.

scholarly journals Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core

2016 ◽  
Vol 7 (3) ◽  
pp. 74-98 ◽  
Author(s):  
Christoph Erath ◽  
Mark A. Taylor ◽  
Ramachandran D. Nair
Keyword(s):  
Spectral Element Method ◽  
Spectral Element ◽  
Tracer Transport ◽  
Third Order ◽  
Dynamical Core ◽  
Community Atmosphere Model ◽  
High Scalability ◽  
Passive Tracers ◽  
Cubed Sphere ◽  
A Current

Abstract In today’s atmospheric numerical modeling, scalable and highly accurate numerical schemes are of particular interest. To address these issues Galerkin schemes, such as the spectral element method, have received more attention in the last decade. They also provide other state-of-the-art capabilities such as improved conservation. However, the tracer transport of hundreds of tracers, e.g., in the chemistry version of the Community Atmosphere Model, is still a performance bottleneck. Therefore, we consider two conservative semi-Lagrangian schemes. Both are designed to be multi-tracer efficient, third order accurate, and allow significantly longer time steps than explicit Eulerian formulations. We address the difficulties arising on the cubed-sphere projection and on parallel computers and show the high scalability of our approach. Additionally, we use the two schemes for the transport of passive tracers in a dynamical core and compare our results with a current spectral element tracer transport advection used by the High-Order Method Modeling Environment.

scholarly journals Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T33-T43
Author(s):  
Chao Lyu ◽  
Yann Capdeville ◽  
Liang Zhao
Keyword(s):  
Wave Equation ◽  
Spectral Element Method ◽  
Computing Time ◽  
Spectral Element ◽  
Two Dimensions ◽  
Elastic Model ◽  
Time Step ◽  
High Degree ◽  
Very High ◽  
Element Method

The spectral element method (SEM) has gained tremendous popularity within the seismological community to solve the wave equation at all scales. Classic SEM applications mostly rely on degrees 4–8 elements in each tensorial direction. Higher degrees are usually not considered due to two main reasons. First, high degrees imply large elements, which make the meshing of mechanical discontinuities difficult. Second, the SEM’s collocation points cluster toward the edge of the elements with the degree, degrading the time-marching stability criteria and imposing a small time step and a high numerical cost. Recently, the homogenization method has been introduced in seismology. This method can be seen as a preprocessing step before solving the wave equation that smooths out the internal mechanical discontinuities of the elastic model. It releases the meshing constraint and makes use of very high degree elements more attractive. Thus, we address the question of memory and computing time efficiency of very high degree elements in SEM, up to degree 40. Numerical analyses reveal that, for a fixed accuracy, very high degree elements require less computer memory than low-degree elements. With minimum sampling points per minimum wavelength of 2.5, the memory needed for a degree 20 is about a quarter that of the one necessary for a degree 4 in two dimensions and about one-eighth in three dimensions. Moreover, for the SEM codes tested in this work, the computation time with degrees 12–24 can be up to twice faster than the classic degree 4. This makes SEM with very high degrees attractive and competitive for solving the wave equation in many situations.

Unified integro-differential equation for efficient dispersive FDTD simulations

2017 ◽  
Vol 36 (4) ◽  
pp. 1089-1105 ◽  
Author(s):  
Omar Ramadan
Keyword(s):  
Differential Equation ◽  
Time Domain ◽  
Stability Limit ◽  
Model Parameters ◽  
Time Step ◽  
Content Type ◽  
Integro Differential Equation ◽  
Dispersive Model ◽  
Fdtd Simulations ◽  
Dispersive Models

Purpose The purpose of this paper is to derive a unified formulation for incorporating different dispersive models into the explicit and implicit finite difference time domain (FDTD) simulations. Design/methodology/approach In this paper, dispersive integro-differential equation (IDE) FDTD formulation is presented. The resultant IDE is written in the discrete time domain by applying the trapezoidal recursive convolution and central finite differences schemes. In addition, unconditionally stable implicit split-step (SS) FDTD implementation is also discussed. Findings It is found that the time step stability limit of the explicit IDE-FDTD formulation maintains the conventional Courant–Friedrichs–Lewy (CFL) constraint but with additional stability limits related to the dispersive model parameters. In addition, the CFL stability limit can be removed by incorporating the implicit SS scheme into the IDE-FDTD formulation, but this is traded for degradation in the accuracy of the formulation. Research limitations/implications The stability of the explicit FDTD scheme is bounded not only by the CFL limit but also by additional condition related to the dispersive material parameters. In addition, it is observed that implicit JE-IDE FDTD implementation decreases as the time step exceeds the CFL limit. Practical implications Based on the presented formulation, a single dispersive FDTD code can be written for implementing different dispersive models such as Debye, Drude, Lorentz, critical point and the quadratic complex rational function. Originality/value The proposed formulation not only unifies the FDTD implementation of the frequently used dispersive models with the minimal storage requirements but also can be incorporated with the implicit SS scheme to remove the CFL time step stability constraint.

Nonlinear Numerical Prediction of Gas Foil Bearing Stability and Unbalanced Response

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Sébastien Le Lez ◽  
Mihaï Arghir ◽  
Jean Frêne
Keyword(s):  
Dry Friction ◽  
Stability Limit ◽  
Time Step ◽  
Structural Dynamic ◽  
Friction Forces ◽  
Foil Bearings ◽  
Mass Unbalance ◽  
Foil Bearing ◽  
Highly Nonlinear ◽  
The Stability

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared with conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly nonlinear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these nonlinearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full nonlinear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared with rigid ones. Moreover, when friction is introduced, a new level of stability is reached, revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a nonlinear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

scholarly journals Specified dynamics scheme impacts on wave-mean flow dynamics, convection, and tracer transport in CESM2 (WACCM6)

2022 ◽  
Vol 22 (1) ◽  
pp. 197-214
Author(s):  
Nicholas A. Davis ◽  
Patrick Callaghan ◽  
Isla R. Simpson ◽  
Simone Tilmes
Keyword(s):  
Climate Model ◽  
Tracer Transport ◽  
Residual Circulation ◽  
Mean Flow ◽  
Time Step ◽  
Modeling Tools ◽  
Transport Pathways ◽  
Systematic Assessment ◽  
Model Version ◽  
Wide Range

Abstract. Specified dynamics schemes are ubiquitous modeling tools for isolating the roles of dynamics and transport on chemical weather and climate. They typically constrain the circulation of a chemistry–climate model to the circulation in a reanalysis product through linear relaxation. However, recent studies suggest that these schemes create a divergence in chemical climate and the meridional circulation between models and do not accurately reproduce trends in the circulation. In this study we perform a systematic assessment of the specified dynamics scheme in the Community Earth System Model version 2, Whole Atmosphere Community Climate Model version 6 (CESM2 (WACCM6)), which proactively nudges the circulation toward the reference meteorology. Specified dynamics experiments are performed over a wide range of nudging timescales and reference meteorology frequencies, with the model's circulation nudged to its own free-running output – a clean test of the specified dynamics scheme. Errors in the circulation scale robustly and inversely with meteorology frequency and have little dependence on the nudging timescale. However, the circulation strength and errors in tracers, tracer transport, and convective mass flux scale robustly and inversely with the nudging timescale. A 12 to 24 h nudging timescale at the highest possible reference meteorology frequency minimizes errors in tracers, clouds, and the circulation, even up to the practical limit of one reference meteorology update every time step. The residual circulation and eddy mixing integrate tracer errors and accumulate them at the end of their characteristic transport pathways, leading to elevated error in the upper troposphere and lower stratosphere and in the polar stratosphere. Even in the most ideal case, there are non-negligible errors in tracers introduced by the nudging scheme. Future development of more sophisticated nudging schemes may be necessary for further progress.

scholarly journals Element-by-element parallel spectral-element methods for 3-D teleseismic wave modeling

Solid Earth ◽  
2017 ◽  
Vol 8 (5) ◽  
pp. 969-986 ◽  
Author(s):  
Shaolin Liu ◽  
Dinghui Yang ◽  
Xingpeng Dong ◽  
Qiancheng Liu ◽  
Yongchang Zheng
Keyword(s):  
Boundary Condition ◽  
Wave Field ◽  
Scattered Wave ◽  
Spectral Element ◽  
Perfectly Matched Layers ◽  
Time Step ◽  
Absorbing Boundary ◽  
Field Modeling ◽  
Adjoint Tomography ◽  
Gradient Calculation

Abstract. The development of an efficient algorithm for teleseismic wave field modeling is valuable for calculating the gradients of the misfit function (termed misfit gradients) or Fréchet derivatives when the teleseismic waveform is used for adjoint tomography. Here, we introduce an element-by-element parallel spectral-element method (EBE-SEM) for the efficient modeling of teleseismic wave field propagation in a reduced geology model. Under the plane-wave assumption, the frequency–wavenumber (FK) technique is implemented to compute the boundary wave field used to construct the boundary condition of the teleseismic wave incidence. To reduce the memory required for the storage of the boundary wave field for the incidence boundary condition, a strategy is introduced to efficiently store the boundary wave field on the model boundary. The perfectly matched layers absorbing boundary condition (PML ABC) is formulated using the EBE-SEM to absorb the scattered wave field from the model interior. The misfit gradient can easily be constructed in each time step during the calculation of the adjoint wave field. Three synthetic examples demonstrate the validity of the EBE-SEM for use in teleseismic wave field modeling and the misfit gradient calculation.

Analysis of Newmark Explicit Integration Method for Nonlinear Systems

2006 ◽  
Vol 22 (4) ◽  
pp. 321-329 ◽  
Author(s):  
S.-Y. Chang ◽  
Y.-C. Huang ◽  
C.-H. Wang
Keyword(s):  
Nonlinear Systems ◽  
Integration Method ◽  
Stability Limit ◽  
Explicit Method ◽  
Initial Stiffness ◽  
Explicit Integration ◽  
Time Step ◽  
Linear Elastic ◽  
Stiffness Softening ◽  
Theoretical Results

AbstractNumerical properties of the Newmark explicit method in the solution of nonlinear systems are explored. It is found that the upper stability limit is no longer equal to 2 for the Newmark explicit method for nonlinear systems. In fact, it is enlarged for stiffness softening and is reduced for stiffness hardening. Furthermore, its relative period error increases with the increase of the step degree of nonlinearity for a given value of the product of the natural frequency and the time step. It is also verified that the viscous damping determined from an initial stiffness is effective to reduce displacement response in the solution of a nonlinear system as that for solving a linear elastic system. All the theoretical results are confirmed with numerical examples.

scholarly journals Effect of a High-Order Filter on a Cubed-Sphere Spectral Element Dynamical Core

2018 ◽  
Vol 146 (7) ◽  
pp. 2047-2064 ◽  
Author(s):  
Hyun-Gyu Kang ◽  
Hyeong-Bin Cheong
Keyword(s):  
Baroclinic Instability ◽  
Spectral Element ◽  
High Order ◽  
Local Domain ◽  
Step Size ◽  
Time Step ◽  
Dynamical Core ◽  
Time Step Size ◽  
Order Filter ◽  
Cubed Sphere

Abstract A high-order filter for a cubed-sphere spectral element model was implemented in a three-dimensional spectral element dry hydrostatic dynamical core. The dynamical core incorporated hybrid sigma–pressure vertical coordinates and a third-order Runge–Kutta time-differencing method. The global high-order filter and the local-domain high-order filter, requiring numerical operation with a huge sparse global matrix and a locally assembled matrix, respectively, were applied to the prognostic variables, except for surface pressure, at every time step. Performance of the high-order filter was evaluated using the baroclinic instability test and quiescent atmosphere with underlying topography test presented by the Dynamical Core Model Intercomparison Project. It was revealed that both the global and local-domain high-order filters could better control the numerical noise in the noisy circumstances than the explicit diffusion, which is widely used for the spectral element dynamical core. Furthermore, by adopting the high-order filter, the effective resolution of the dynamical core could be increased, without weakening the stability of the dynamical core. Computational efficiency of the high-order filter was demonstrated in terms of both the time step size and the wall-clock time. Because of the nature of an implicit diffusion, the dynamical core employing this filter can take a larger time step size, compared to that using the explicit diffusion. The local-domain high-order filter was computationally more efficient than the global high-order filter, but less efficient than the explicit diffusion.

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