scholarly journals Comparison of Terrain-Following and Cut-Cell Grids Using a Nonhydrostatic Model

2016 ◽  
Vol 144 (6) ◽  
pp. 2085-2099 ◽  
Author(s):  
James Shaw ◽  
Hilary Weller
Keyword(s):  
Velocity Field ◽  
Gravity Waves ◽  
Initial Conditions ◽  
Implicit Time ◽  
Test Results ◽  
Time Step ◽  
Induced Gravity ◽  
Time Stepping ◽  
Cut Cell ◽  
Terrain Following

Abstract Terrain-following coordinates are widely used in operational models but the cut-cell method has been proposed as an alternative that can more accurately represent atmospheric dynamics over steep orography. Because the type of grid is usually chosen during model implementation, it becomes necessary to use different models to compare the accuracy of different grids. In contrast, here a C-grid finite-volume model enables a like-for-like comparison of terrain-following and cut-cell grids. A series of standard two-dimensional tests using idealized terrain are performed: tracer advection in a prescribed horizontal velocity field, a test starting from resting initial conditions, and orographically induced gravity waves described by nonhydrostatic dynamics. In addition, three new tests are formulated: a more challenging resting atmosphere case, and two new advection tests having a velocity field that is everywhere tangential to the terrain-following coordinate surfaces. These new tests present a challenge on cut-cell grids. The results of the advection tests demonstrate that accuracy depends primarily upon alignment of the flow with the grid rather than grid orthogonality. A resting atmosphere is well maintained on all grids. In the gravity waves test, results on all grids are in good agreement with existing results from the literature, although terrain-following velocity fields lead to errors on cut-cell grids. Because of semi-implicit time stepping and an upwind-biased, explicit advection scheme, there are no time step restrictions associated with small cut cells. In contradiction to other studies, no significant advantages of cut cells or smoothed coordinates are found.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

Nonlinear-wave effects on fixed and floating bodies

1982 ◽  
Vol 120 ◽  
pp. 267-281 ◽  
Author(s):  
Michael De St Q. Isaacson
Keyword(s):  
Initial Conditions ◽  
Integral Equation Method ◽  
Ocean Waves ◽  
Floating Bodies ◽  
Time Step ◽  
Fluid Forces ◽  
Time Stepping ◽  
Transient Problem ◽  
Wave Effects ◽  
Incident Waves

A numerical method for calculating the interaction of steep (nonlinear) ocean waves with large fixed or floating structures of arbitrary shape is described. The interaction is treated as a transient problem with known initial conditions corresponding to still water in the vicinity of the structure and a prescribed incident waveform approaching it. The development of the flow, together with the associated fluid forces and structural motions, are obtained by a time-stepping procedure in which the flow at each time step is calculated by an integral-equation method based on Green's theorem. A few results are presented for two reference situations and these serve to illustrate the effects of nonlinearities in the incident waves.

scholarly journals Efficient extrapolation methods for electro- and magnetoquasistatic field simulations

2003 ◽  
Vol 1 ◽  
pp. 81-86 ◽  
Author(s):  
M. Clemens ◽  
M. Wilke ◽  
T. Weiland
Keyword(s):  
Taylor Series Expansion ◽  
Iterative Solvers ◽  
Gradient Algorithm ◽  
Implicit Time ◽  
Preconditioned Conjugate Gradient ◽  
Test Problems ◽  
Time Step ◽  
Time Stepping ◽  
Multi Stage ◽  
Runge Kutta

Abstract. In magneto- and electroquasi-static time domain simulations with implicit time stepping schemes the iterative solvers applied to the large sparse (non-)linear systems of equations are observed to converge faster if more accurate start solutions are available. Different extrapolation techniques for such new time step solutions are compared in combination with the preconditioned conjugate gradient algorithm. Simple extrapolation schemes based on Taylor series expansion are used as well as schemes derived especially for multi-stage implicit Runge-Kutta time stepping methods. With several initial guesses available, a new subspace projection extrapolation technique is proven to produce an optimal initial value vector. Numerical tests show the resulting improvements in terms of computational efficiency for several test problems. In quasistatischen elektromagnetischen Zeitbereichsimulationen mit impliziten Zeitschrittverfahren zeigt sich, dass die iterativen Lösungsverfahren für die großen dünnbesetzten (nicht-)linearen Gleichungssysteme schneller konvergieren, wenn genauere Startlösungen vorgegeben werden. Verschiedene Extrapolationstechniken werden für jeweils neue Zeitschrittlösungen in Verbindung mit dem präkonditionierten Konjugierte Gradientenverfahren vorgestellt. Einfache Extrapolationsverfahren basierend auf Taylorreihenentwicklungen werden ebenso benutzt wie speziell für mehrstufige implizite Runge-Kutta-Verfahren entwickelte Verfahren. Sind verschiedene Startlösungen verfügbar, so erlaubt ein neues Unterraum-Projektion- Extrapolationsverfahren die Konstruktion eines optimalen neuen Startvektors. Numerische Tests zeigen die aus diesen Verfahren resultierenden Verbesserungen der numerischen Effizienz.

Generation of inertia-gravity waves in idealized baroclinic-wave life cycles: Explicit vs. semi-implicit time stepping in a finite-volume solver for the pseudo-incompressible equations

2020 ◽  
Author(s):  
Fabienne Schmid ◽  
Elena Gagarina ◽  
Rupert Klein ◽  
Ulrich Achatz
Keyword(s):  
Finite Volume ◽  
Spontaneous Emission ◽  
Gravity Waves ◽  
Prediction Models ◽  
Weather Prediction ◽  
Life Cycles ◽  
Staggered Grid ◽  
Implicit Time ◽  
Baroclinic Wave ◽  
Time Stepping

<div> <div> <div> <p>Inertia–gravity waves (IGWs) emitted from jets and fronts are ubiquitous in the atmosphere and have a significant impact on atmospheric processes (Plougonven and Zhang, 2014). Since the mechanism responsible for the spontaneous emission of IGWs during the evolution of an initially balanced flow remain poorly understood, their representation in numerical weather prediction models is challenging (de la Cámara and Lott, 2015). Better understanding of this IGW source mechanism based on idealized numerical simulations is crucial to improve the accuracy of the forecasts. In this study, idealized baroclinic-wave life cycle experiments on the f-plane are performed to investigate spontaneous emission, using a finite-volume solver for the pseudo-incompressible equations (Rieper et al., 2013). In particular, the implementation of a semi-implicit time stepping scheme, along the lines of Smolarkiewicz and Margolin (1997) and Benaccio and Klein (2019), but adjusted to our staggered grid, permits longer simulation runs with much larger domains. A novelty is the implementation of a simple Newtonian heating function based on Held and Suarez (1994), which is used for forcing a baroclinically unstable temperature profile and allows the background state to vary in time (O’Neill and Klein, 2014). The results of the model with semi-implicit time stepping scheme will be documented and compared to an explicit Runge-Kutta scheme. The analysis may serve as a basis for the development and validation of a parameterization scheme for GWs emitted from jets and fronts.</p> <p>References:</p> <p>Benaccio, T., and R. Klein, 2019: A semi-implicit compressible model for atmospheric flows with seamless access to soundproof and hydrostatic dynamics. Mon. Wea. Rev., 147, 4221-4240.<br>de la Cámara, A., and F. Lott, 2015: A parameterization of gravity waves emitted by fronts and jets. Geophys. Res. Lett., 42, 2071-2078.<br>Held, I.M., and M.J. Suarez, 1994: A Proposal for the Intercomparison of the Dynamical Cores of Atmospheric General Circulation Models. Bull. Amer. Meteor. Soc., 75, 1825-1830.<br>O’Neill, W.P., and R. Klein, 2014: A moist pseudo-incompressible model. Atmos. Res., 142, 133-141. Plougonven R., and F. Zhang, 2014: Internal gravity waves from atmospheric jets and fronts. Rev. Geophys., 52, 33-76.<br>Rieper, F., Hickel, S., and U. Achatz, 2013: A conservative integration of the pseudo-incompressible equations with implicit turbulence parameterization. Mon. Wea. Rev., 141, 861-886. Smolarkiewicz, P.K., and L.G. Margolin, 1997: On forward-in-time differencing for fluids: an Eulerian/semi-Langrangian nonhydrostatic model for stratified flows. Atmosphere-Ocean, 35, 127- 152.</p> </div> </div> </div>

Well-balanced compressible cut-cell simulation of atmospheric flow

2009 ◽  
Vol 367 (1907) ◽  
pp. 4559-4575 ◽  
Author(s):  
R. Klein ◽  
K. R. Bates ◽  
N. Nikiforakis
Keyword(s):  
Attractive Alternative ◽  
Test Problems ◽  
Time Step ◽  
Atmospheric Flow ◽  
Flow Equations ◽  
Cut Cell ◽  
Flux Approximation ◽  
Terrain Following ◽  
Lee Wave ◽  
Cell Simulation

Cut-cell meshes present an attractive alternative to terrain-following coordinates for the representation of topography within atmospheric flow simulations, particularly in regions of steep topographic gradients. In this paper, we present an explicit two-dimensional method for the numerical solution on such meshes of atmospheric flow equations including gravitational sources. This method is fully conservative and allows for time steps determined by the regular grid spacing, avoiding potential stability issues due to arbitrarily small boundary cells. We believe that the scheme is unique in that it is developed within a dimensionally split framework, in which each coordinate direction in the flow is solved independently at each time step. Other notable features of the scheme are: (i) its conceptual and practical simplicity, (ii) its flexibility with regard to the one-dimensional flux approximation scheme employed, and (iii) the well-balancing of the gravitational sources allowing for stable simulation of near-hydrostatic flows. The presented method is applied to a selection of test problems including buoyant bubble rise interacting with geometry and lee-wave generation due to topography.

scholarly journals A Numerical Method Based on Leapfrog and a Fourth-Order Implicit Time Filter

2014 ◽  
Vol 142 (7) ◽  
pp. 2545-2560 ◽  
Author(s):  
Mohamed Moustaoui ◽  
Alex Mahalov ◽  
Eric J. Kostelich
Keyword(s):  
Fourth Order ◽  
Climate Modeling ◽  
Implicit Time ◽  
Time Step ◽  
Third Order ◽  
Time Stepping ◽  
Formal Stability ◽  
Numerical Instabilities ◽  
Potential Alternative ◽  
Leapfrog Scheme

Abstract A time-stepping scheme is proposed. It is based on the leapfrog method and a fourth-order time filter. The scheme requires only one evaluation per time step and uses an implicit filter, but the effort needed to implement it in an explicit manner is trivial. Comparative tests demonstrate that the proposed scheme produces numerical approximations that are more stable and highly accurate compared to the standard Robert–Asselin (RA) and the Robert–Asselin–Williams (RAW) filtered leapfrog scheme, even though both methods use filter coefficients that are tuned such that the 2Δt modes are damped at the same rate. Formal stability analysis demonstrates that the proposed method generates amplitude errors of O[(Δt)4], implying third-order accuracy. This contrasts with the O[(Δt)2] errors produced by the standard RA and RAW filtered leapfrog. A second scheme that produces amplitude errors of O[(Δt)6] is also presented. The proposed scheme is found to do well at controlling numerical instabilities arising in the diffusion equation and in nonlinear computations using Lorenz’s system and the global shallow-water spectral model. In addition to noticeably improving the resolution of the physical modes, the proposed method is simple to implement and has a wider region of stability compared to the existing time-filtered leapfrog schemes. This makes the proposed method a potential alternative for use in atmospheric, oceanic, and climate modeling.

scholarly journals Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

2021 ◽  
Author(s):  
Jan Ackmann ◽  
Peter Düben ◽  
Tim Palmer ◽  
Piotr Smolarkiewicz
Keyword(s):  
Machine Learning ◽  
Shallow Water ◽  
Water Model ◽  
Implicit Time ◽  
Learning Approaches ◽  
Shallow Water Model ◽  
Time Step ◽  
Linear Solvers ◽  
Linear Solver ◽  
Time Stepping

<p>Semi-implicit grid-point models for the atmosphere and the ocean require linear solvers that are working efficiently on modern supercomputers. The huge advantage of the semi-implicit time-stepping approach is that it enables large model time-steps. This however comes at the cost of having to solve a computationally demanding linear problem each model time-step to obtain an update to the model’s pressure/fluid-thickness field. In this study, we investigate whether machine learning approaches can be used to increase the efficiency of the linear solver.</p><p>Our machine learning approach aims at replacing a key component of the linear solver—the preconditioner. In the preconditioner an approximate matrix inversion is performed whose quality largely defines the linear solver’s performance. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding complexity and machine learning design choices.</p><p>Several machine learning methods of different complexity from simple linear regression to deep feed-forward neural networks are used to learn the optimal preconditioner for a shallow-water model with semi-implicit time-stepping. The shallow-water model is specifically designed to be conceptually similar to more complex atmosphere models. The machine-learning preconditioner is competitive with a conventional preconditioner and provides good results even if it is used outside of the dynamical range of the training dataset.</p>

An Implicit Time-Stepping Method for Quasi-Rigid Multibody Systems With Intermittent Contact

2007 ◽  
Author(s):  
Nilanjan Chakraborty ◽  
Stephen Berard ◽  
Srinivas Akella ◽  
Jeff Trinkle
Keyword(s):  
Complementarity Problem ◽  
The Body ◽  
Contact Forces ◽  
Implicit Time ◽  
Discrete Problem ◽  
Time Step ◽  
Normal Contact ◽  
Time Stepping ◽  
Intermittent Contact ◽  
Compliant Layer

We recently developed a time-stepping method for simulating rigid multi-body systems with intermittent contact that is implicit in the geometric information [1]. In this paper, we extend this formulation to quasi-rigid or locally compliant objects, i.e., objects with a rigid core surrounded by a compliant layer, similar to Song et al. [2]. The difference in our compliance model from existing quasi-rigid models is that, based on physical motivations, we assume the compliant layer has a maximum possible normal deflection beyond which it acts as a rigid body. Therefore, we use an extension of the Kelvin-Voigt (i.e. linear spring-damper) model for obtaining the normal contact forces by incorporating the thickness of the compliant layer explicitly in the contact model. We use the Kelvin-Voigt model for the tangential forces and assume that the contact forces and moment satisfy an ellipsoidal friction law. We model each object as an intersection of convex inequalities and write the contact constraint as a complementarity constraint between the contact force and a distance function dependent on the closest points and the local deformation of the body. The closest points satisfy a system of nonlinear algebraic equations and the resultant continuous model is a Differential Complementarity Problem (DCP). This enables us to formulate a geometrically implicit time-stepping scheme for solving the DCP which is more accurate than a geometrically explicit scheme. The discrete problem to be solved at each time-step is a mixed nonlinear complementarity problem.

scholarly journals An adaptive BDF2 implicit time-stepping method for the phase field crystal model

2020 ◽  
Author(s):  
Hong-lin Liao ◽  
Bingquan Ji ◽  
Luming Zhang
Keyword(s):  
Error Estimate ◽  
Phase Field ◽  
Implicit Time ◽  
Time Step ◽  
Time Stepping ◽  
Nonlinear Parabolic ◽  
Crystal Model ◽  
Discrete Orthogonal ◽  
Convolution Kernels ◽  
Phase Field Crystal

Abstract An adaptive BDF2 implicit time-stepping method is analyzed for the phase field crystal model. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels when the time-step ratios satisfy $r_k:=\tau _k/\tau _{k-1}<3.561$, which is the zero-stability restriction of the variable-step BDF2 scheme for ordinary differential equations. With the help of discrete orthogonal convolution kernels and corresponding convolution inequalities, an optimal $L^2$ norm error estimate is established under the weak step-ratio restriction $0<r_k<3.561$ to ensure energy stability. As far as we know, this is the first time that such an error estimate is theoretically proved for a nonlinear parabolic equation. Based on tests on random temporal meshes an effective adaptive time-stepping strategy is suggested to efficiently capture the multi-scale behavior and accelerate the numerical simulations.

Testing of a Cell-Integrated Semi-Lagrangian Semi-Implicit Nonhydrostatic Atmospheric Solver (CSLAM-NH) with Idealized Orography

2015 ◽  
Vol 143 (4) ◽  
pp. 1382-1398
Author(s):  
May Wong ◽  
William C. Skamarock ◽  
Peter H. Lauritzen ◽  
Joseph B. Klemp ◽  
Roland B. Stull
Keyword(s):  
Large Time ◽  
Cloud Formation ◽  
Implicit Time ◽  
Mountain Waves ◽  
Time Stepping ◽  
Terrain Following ◽  
A Cell ◽  
Passive Tracers ◽  
The Stability ◽  
Orographic Cloud

Abstract A recently developed cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic atmospheric solver that uses the conservative semi-Lagrangian multitracer (CSLAM) transport scheme is extended to include orographic influences. With the introduction of a new semi-implicit CISL discretization of the continuity equation, the nonhydrostatic solver, called CSLAM-NH, has been shown to ensure inherently conservative and numerically consistent transport of air mass and other scalar variables, such as moisture and passive tracers. The extended CSLAM-NH presented here includes two main modifications: transformation of the equation set to a terrain-following height coordinate to incorporate orography and an iterative centered-implicit time-stepping scheme to enhance the stability of the scheme associated with gravity wave propagation at large time steps. CSLAM-NH is tested for a suite of idealized 2D flows, including linear mountain waves (dry), a downslope windstorm (dry), and orographic cloud formation.

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