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.

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.

A Fast Computational Method for Particle Dynamics in Dielectrophoretic Lab-on-Chip Systems

2007 ◽  
Author(s):  
Indranil Chowdhury ◽  
Vikram Jandhyala ◽  
John D. Rockway
Keyword(s):  
Field Distribution ◽  
Iterative Solvers ◽  
Computational Method ◽  
Microfluidic Channels ◽  
Computational Domain ◽  
Time Step ◽  
Lab On Chip ◽  
Time Stepping ◽  
Fast Solver ◽  
On Chip

An accelerated boundary element method (BEM) is proposed for predicting the motion of bio-particles under combined electromagnetic and fluidic force fields. Many Lab-on-chip (LoC) designs are based on dielectrophoretic (DEP) manipulation of polarized species inside microfluidic channels. The BEM approach presented here relies entirely on modeling the surface of the computational domain, significantly reducing the number of unknowns when compared to volume-based methods. Additionally, the need for re-meshing the whole domain at each time-step of particle movement is prevented. A coupled circuit-EM formulation is presented for accurate prediction of dielectophoretic field distribution due to on-chip electrodes. This allows the circuit control of the resulting electromagnetic fields. Next, BEM formulations for predicting DEP and fluidic traction forces on arbitrarily shaped bio-particles are presented. EM fields produced by the electrodes induce the DEP forces, while the fluid flow is driven by a pressure gradient across the channel. The resultant motion of the subjected particles is studied using a simple time-stepping algorithm. The algorithm has a time complexity of O(N3), where N is the number of unknowns), which leads to a large bottleneck during simulation of each time step. This problem is addressed by implementing oct-tree based O(N) multilevel iterative solvers. The methodology is used to study the field distribution due to distributed electrode systems and particle motion in fluidic channels. Evidence of O(N) behavior of the fast solver is presented. The resulting simulator can be used to study complicated distributed structures and explore new LoC design ideas.

scholarly journals A TIME ACCURATE PSEUDO-WAVELET SCHEME FOR TWO-DIMENSIONAL TURBULENCE

2005 ◽  
Vol 03 (04) ◽  
pp. 587-599 ◽  
Author(s):  
B. V. RATHISH KUMAR ◽  
MANI MEHRA
Keyword(s):  
Galerkin Method ◽  
Taylor Series ◽  
Stokes Equations ◽  
Taylor Series Expansion ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Time Stepping ◽  
Taylor Galerkin Method

In this paper, we propose a wavelet-Taylor–Galerkin method for solving the two-dimensional Navier–Stokes equations. The discretization in time is performed before the spatial discretization by introducing second-order generalization of the standard time stepping schemes with the help of Taylor series expansion in time step. Wavelet-Taylor–Galerkin schemes taking advantage of the wavelet bases capabilities to compress both functions and operators are presented. Results for two-dimensional turbulence are shown.

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>

EARLY EXPERIENCES WITH THE 360TF IBM BLUE GENE/L PLATFORM

2008 ◽  
Vol 05 (02) ◽  
pp. 237-253 ◽  
Author(s):  
G. BHANOT ◽  
J. M. DENNIS ◽  
J. EDWARDS ◽  
W. GRABOWSKI ◽  
M. GUPTA ◽  
...  
Keyword(s):  
Test Problem ◽  
Time Integration ◽  
Atmospheric Model ◽  
Horizontal Resolution ◽  
Implicit Time ◽  
Preconditioned Conjugate Gradient ◽  
Time Step ◽  
Climate System Model ◽  
Vertical Coordinate ◽  
Blue Gene

The High Order Method Modeling Environment is a scalable, spectral-element-based prototype for the Community Atmospheric Model component of the Community Climate System Model. The 3D moist primitive equations are solved on the cubed sphere with a hybrid pressure η vertical coordinate using an Emanuel convective parametrization for moist processes. Semi-implicit time integration, based on a preconditioned conjugate gradient solver, circumvents the time step restrictions associated with gravity waves. Benchmarks for two standard tests problems at 10 km horizontal resolution have been run on Blue Gene/L. Results obtained on a 32-rack Blue Gene/L system (65,536 processors, 183.5-teraflop peak) show sustained performance of 8.0 teraflops on 32,768 processors for the moist Held–Suarez test problem in coprocessor mode and 11.3 teraflops on 32,768 processors for the aquaplanet test problem, running in virtual node mode.

scholarly journals Irksome: Automating Runge–Kutta Time-stepping for Finite Element Methods

2021 ◽  
Vol 47 (4) ◽  
pp. 1-26
Author(s):  
Patrick E. Farrell ◽  
Robert C. Kirby ◽  
Jorge Marchena-Menéndez
Keyword(s):  
Finite Element ◽  
Algebraic System ◽  
Efficient Solution ◽  
High Order Accuracy ◽  
Time Step ◽  
Order Accuracy ◽  
Numerical Examples ◽  
Time Stepping ◽  
Runge Kutta ◽  
High Level

While implicit Runge–Kutta (RK) methods possess high order accuracy and important stability properties, implementation difficulties and the high expense of solving the coupled algebraic system at each time step are frequently cited as impediments. We present Irksome , a high-level library for manipulating UFL (Unified Form Language) expressions of semidiscrete variational forms to obtain UFL expressions for the coupled Runge–Kutta stage equations at each time step. Irksome works with the Firedrake package to enable the efficient solution of the resulting coupled algebraic systems. Numerical examples confirm the efficacy of the software and our solver techniques for various problems.

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 Adaptive Optimal -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-25
Author(s):  
Jui-Ling Yu
Keyword(s):  
Method Of Lines ◽  
Reaction Diffusion ◽  
Stability And Convergence ◽  
Optimal Time ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Application Problem ◽  
Runge Kutta

We present a class of numerical methods for the reaction-diffusion-chemotaxis system which is significant for biological and chemistry pattern formation problems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. Along with the implementation of the method of lines, implicit or semi-implicit schemes are typical time stepping solvers to reduce the effect on time step constrains due to the stability condition. However, these two schemes are usually difficult to employ. In this paper, we propose an adaptive optimal time stepping strategy for the explicit -stage Runge-Kutta method to solve reaction-diffusion-chemotaxis systems. Instead of relying on empirical approaches to control the time step size, variable time step sizes are given explicitly. Yet, theorems about stability and convergence of the algorithm are provided in analyzing robustness and efficiency. Numerical experiment results on a testing problem and a real application problem are shown.

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