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.

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].

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.

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.

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.

AN ITERATIVE TIME-STEPPING METHOD FOR SOLVING FIRST-ORDER TIME DEPENDENT PROBLEMS AND ITS APPLICATION TO THE WAVE EQUATION

2000 ◽  
Vol 08 (01) ◽  
pp. 241-255 ◽  
Author(s):  
GÉZA SERIANI
Keyword(s):  
Wave Equation ◽  
Time Integration ◽  
Spectral Element ◽  
Implicit Time ◽  
Time Step ◽  
Time Stepping ◽  
Point Scheme ◽  
Time Marching ◽  
Matrix Vector ◽  
Time Dependent Problems

Equations describing dynamic problems, after spatial discretization by using the finite element or spectral element method, lead to solve large systems of ODE in time. A family of new time integration algorithms based on an iterative time-stepping (ITS) approach is proposed for solving these systems. The method is developed for first-and second-order differential equations, and applied, in particular, to wave equation. It is an implicit time marching method in which, at each time-step, the solution is computed by a fixed-point scheme. The analysis show that the method is accurate, unconditionally stable and that it allows for efficient and parallel implementations because no matrix inversion is required and only matrix-vector multiplications and vector scaling operations are involved.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

scholarly journals Perturbative linearization of super-Yang-Mills theories in general gauges

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Hannes Malcha ◽  
Hermann Nicolai
Keyword(s):  
Large Class ◽  
Fourth Order ◽  
Landau Gauge ◽  
Third Order ◽  
Yang Mills ◽  
Free Theory ◽  
Non Linear ◽  
Functional Measure ◽  
Non Local ◽  
Jacobi Determinant

Abstract Supersymmetric Yang-Mills theories can be characterized by a non-local and non-linear transformation of the bosonic fields (Nicolai map) mapping the interacting functional measure to that of a free theory, such that the Jacobi determinant of the transformation equals the product of the fermionic determinants obtained by integrating out the gauginos and ghosts at least on the gauge hypersurface. While this transformation has been known so far only for the Landau gauge and to third order in the Yang-Mills coupling, we here extend the construction to a large class of (possibly non-linear and non-local) gauges, and exhibit the conditions for all statements to remain valid off the gauge hypersurface. Finally, we present explicit results to second order in the axial gauge and to fourth order in the Landau gauge.

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