MHDSTS: a new explicit numerical scheme for simulations of partially ionised solar plasma

The interaction of plasma with magnetic field in the partially ionised solar atmosphere is frequently modelled via a single-fluid approximation, which is valid for the case of a strongly coupled collisional media, such as solar photosphere and low chromosphere. Under the single-fluid formalism the main non-ideal effects are described by a series of extra terms in the generalised induction equation and in the energy conservation equation. These effects are: Ohmic diffusion, ambipolar diffusion, the Hall effect, and the Biermann battery effect. From the point of view of the numerical solution of the single-fluid equations, when ambipolar diffusion or Hall effects dominate can introduce severe restrictions on the integration time step and can compromise the stability of the numerical scheme. In this paper we introduce two numerical schemes to overcome those limitations. The first of them is known as super time-stepping (STS) and it is designed to overcome the limitations imposed when the ambipolar diffusion term is dominant. The second scheme is called the Hall diffusion scheme (HDS) and it is used when the Hall term becomes dominant. These two numerical techniques can be used together by applying Strang operator splitting. This paper describes the implementation of the STS and HDS schemes in the single-fluid code MANCHA3D. The validation for each of these schemes is provided by comparing the analytical solution with the numerical one for a suite of numerical tests.

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Neimark–Sacker bifurcations in a non-standard numerical scheme for a class of positivity-preserving ODEs

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2006.1724 ◽

2006 ◽

Vol 462 (2074) ◽

pp. 3167-3184 ◽

Cited By ~ 11

Author(s):

Murray E Alexander ◽

Arthur R Summers ◽

Seyed M Moghadas

Keyword(s):

Hopf Bifurcation ◽

Numerical Scheme ◽

Integration Time ◽

Bifurcation Parameter ◽

Two Dimensional ◽

Time Step ◽

Positivity Preserving ◽

Lyapunov Coefficient ◽

Theoretical Results ◽

General Method

We discuss the nature of Neimark–Sacker bifurcations occurring in a non-standard numerical scheme, for a class of positivity-preserving systems of ordinary differential equations (ODEs) which undergoes a corresponding Hopf bifurcation. Extending previous work (Alexander & Moghadas 2005 a Electron. J. Diff. Eqn. Conf . 12 , 9–19), it is shown that the type of Neimark–Sacker bifurcation (supercritical or subcritical) may be affected by the integration time-step . The general form of the scheme in the vicinity of a fixed point is given, from which the expression for the first Lyapunov coefficient for two-dimensional systems, valid for arbitrary time-step, is explicitly derived. The analysis shows that this coefficient undergoes an shift with respect to the corresponding coefficient of the original ODE. This could lead to a type of bifurcation which differs from the corresponding Hopf bifurcation in the ODE, due to changes in the sign of the first Lyapunov coefficient as varies. This is especially problematic in the vicinity of certain types of degenerate Hopf bifurcation, at which this coefficient may vanish. We also present a general method to eliminate the possible shift in the bifurcation parameter of the scheme; however, the first Lyapunov coefficient may still be subjected to an shift, leading to a possibly erroneous type of bifurcation. Examples are given to illustrate the theoretical results of the paper with applications to mathematical biology.

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An Efficient Multi-Scale Local Binary Fitting-Based Level Set Method for Inhomogeneous Image Segmentation

Journal of Sensors ◽

10.1155/2018/6269214 ◽

2018 ◽

Vol 2018 ◽

pp. 1-17

Author(s):

Dengwei Wang

Keyword(s):

Image Segmentation ◽

Level Set ◽

Operator Splitting ◽

Reaction Diffusion ◽

Initial Contour ◽

Time Step ◽

Detection Mechanism ◽

Diffusion Term ◽

Implementation Phase ◽

Level Set Function

An efficient level set model based on multiscale local binary fitting (MLBF) is proposed for image segmentation. By introducing multiscale idea into the LBF model, the proposed MLBF model can effectively and efficiently segment images with intensity inhomogeneity. In addition, by adding a reaction diffusion term into the level set evolution (LSE) equation, the regularization of the level set function (LSF) can be achieved, thus completely eliminating the time-consuming reinitialization process. In the implementation phase, in order to greatly improve the efficiency of the numerical solution of the level set segmentation model, we introduce three strategies: The first is the additive operator splitting (AOS) solver which is used for breaking the restrictions on time step; the second is the salient target detection mechanism which is used to achieve full automatic initialization of the LSE process; the third is the sparse filed method (SFM) which is used to restrict the groups of pixels that need to be updated in a small strip region. Under the combined effect of these three strategies, the proposed model achieves very high execution efficiency in the following aspects: contour location accuracy, speed of evolution convergence, robustness against initial contour position, and robustness against noise interference.

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A Fast Explicit Scheme for Solving MHD Equations with Ambipolar Diffusion

Proceedings of the International Astronomical Union ◽

10.1017/s1743921311000731 ◽

2010 ◽

Vol 6 (S270) ◽

pp. 415-419

Author(s):

Jongsoo Kim

Keyword(s):

Small Time ◽

Ambipolar Diffusion ◽

Mhd Equations ◽

Time Step ◽

Good Test ◽

Diffusion Term ◽

Spatial Gradients ◽

Ideal Mhd ◽

Induction Equation ◽

The Ideal

AbstractWe developed a fast numerical scheme for solving ambipolar diffusion MHD equations with the strong coupling approximation, which can be written as the ideal MHD equations with an additional ambipolar diffusion term in the induction equation. The mass, momentum, magnetic fluxes due to the ideal MHD equations can be easily calculated by any Godunov-type schemes. Additional magnetic fluxes due to the ambipolar diffusion term are added in the magnetic fluxes, because of two same spatial gradients operated on the advection fluxes and the ambipolar diffusion term. In this way, we easily kept divergence-free magnetic fields using the constraint transport scheme. In order to overcome a small time step imposed by ambipolar diffusion, we used the super time stepping method. The resultant scheme is fast and robust enough to do the long term evolution of star formation simulations. We also proposed that the decay of alfen by ambipolar diffusion be a good test problem for our codes.

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The Influence of an Integration Time Step on Dynamic Calculation of a Vehicle-Track-Bridge under High-Speed Railway

Mathematics ◽

10.3390/math9040431 ◽

2021 ◽

Vol 9 (4) ◽

pp. 431

Author(s):

Junjie Ye ◽

Hao Sun

Keyword(s):

High Speed ◽

Coupled Model ◽

Short Wave ◽

Integration Time ◽

Vertical Acceleration ◽

Dynamic Calculation ◽

Time Step ◽

High Speed Railway ◽

Track Irregularity ◽

Track Bridge

In order to study the influence of an integration time step on dynamic calculation of a vehicle-track-bridge under high-speed railway, a vehicle-track-bridge (VTB) coupled model is established. The influence of the integration time step on calculation accuracy and calculation stability under different speeds or different track regularity states is studied. The influence of the track irregularity on the integration time step is further analyzed by using the spectral characteristic of sensitive wavelength. According to the results, the disparity among the effect of the integration time step on the calculation accuracy of the VTB coupled model at different speeds is very small. Higher speed requires a smaller integration time step to keep the calculation results stable. The effect of the integration time step on the calculation stability of the maximum vertical acceleration of each component at different speeds is somewhat different, and the mechanism of the effect of the integration time step on the calculation stability of the vehicle-track-bridge coupled system is that corresponding displacement at the integration time step is different. The calculation deviation of the maximum vertical acceleration of the car body, wheel-sets and bridge under the track short wave irregularity state are greatly increased compared with that without track irregularity. The maximum vertical acceleration of wheel-sets, rails, track slabs and the bridge under the track short wave irregularity state all show a significant declining trend. The larger the vibration frequency is, the smaller the range of integration time step is for dynamic calculation.

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Dynamic Co-Simulation Methods for Combined Transmission-Distribution System With Integration Time Step Impact on Convergence

IEEE Transactions on Power Systems ◽

10.1109/tpwrs.2018.2874807 ◽

2019 ◽

Vol 34 (2) ◽

pp. 1171-1181 ◽

Cited By ~ 7

Author(s):

Ramakrishnan Venkatraman ◽

Siddhartha Kumar Khaitan ◽

Venkataramana Ajjarapu

Keyword(s):

Distribution System ◽

Integration Time ◽

Simulation Methods ◽

Time Step

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IL-GLOBO (1.0) – integrated Lagrangian particle model and Eulerian general circulation model GLOBO: development of the vertical diffusion module

Geoscientific Model Development ◽

10.5194/gmd-7-2181-2014 ◽

2014 ◽

Vol 7 (5) ◽

pp. 2181-2191 ◽

Cited By ~ 4

Author(s):

D. Rossi ◽

A. Maurizi

Keyword(s):

General Circulation Model ◽

General Circulation ◽

Spline Interpolation ◽

Circulation Model ◽

Linear Interpolation ◽

Computational Time ◽

Integration Time ◽

Particle Model ◽

Vertical Diffusion ◽

Time Step

Abstract. The development and validation of the vertical diffusion module of IL-GLOBO, a Lagrangian transport model coupled online with the Eulerian general circulation model GLOBO, is described. The module simulates the effects of turbulence on particle motion by means of a Lagrangian stochastic model (LSM) consistently with the turbulent diffusion equation used in GLOBO. The implemented LSM integrates particle trajectories, using the native σ-hybrid coordinates of the Eulerian component, and fulfils the well-mixed condition (WMC) in the general case of a variable density profile. The module is validated through a series of 1-D offline numerical experiments by assessing its accuracy in maintaining an initially well-mixed distribution in the vertical. A dynamical time-step selection algorithm with constraints related to the shape of the diffusion coefficient profile is developed and discussed. Finally, the skills of a linear interpolation and a modified Akima spline interpolation method are compared, showing that both satisfy the WMC with significant differences in computational time. A preliminary run of the fully integrated 3-D model confirms the result only for the Akima interpolation scheme while the linear interpolation does not satisfy the WMC with a reasonable choice of the minimum integration time step.

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Splitting-particle methods for structured population models: Convergence and applications

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202514500183 ◽

2014 ◽

Vol 24 (11) ◽

pp. 2171-2197 ◽

Cited By ~ 13

Author(s):

J. A. Carrillo ◽

P. Gwiazda ◽

A. Ulikowska

Keyword(s):

Numerical Scheme ◽

Order Of Convergence ◽

Operator Splitting ◽

Population Models ◽

Particle Simulation ◽

Particle Methods ◽

Test Cases ◽

Structured Population Models ◽

Convergence Error ◽

Structured Population

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

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Thermal modeling of hydrogen storage by absorption in a magnesium hydrides tank

International Journal for Simulation and Multidisciplinary Design Optimization ◽

10.1051/smdo/2014002 ◽

2014 ◽

Vol 5 ◽

pp. A21 ◽

Cited By ~ 3

Author(s):

Karim Lahmer ◽

Rachid Bessaïh ◽

Angel Scipioni ◽

Mohammed El Ganaoui

Keyword(s):

Hydrogen Absorption ◽

Numerical Scheme ◽

Computational Time ◽

Governing Equations ◽

Time Step ◽

Kinetic Reaction ◽

Fully Implicit ◽

Temporal Profiles ◽

Tank Geometry ◽

Reaction Equations

This paper summarizes numerical results of hydrogen absorption simulated in an axisymmetric tank geometry containing magnesium hydride heated to 300 °C and at moderate storage pressure 1 MPa. The governing equations are solved with a fully implicit finite volume numerical scheme used by a commercial software FLUENT. The effect of the different kinetic reaction equations modeling hydrogen absorption was studied by the introduction of a specific subroutine at each time step in order to consider which one will provide results close to available experimental results. Spatial and temporal profiles of temperature and concentration in hydride bed are plotted. Results show that suitable method for our two-dimensional study is a CV-2D technique because it generates the smallest error especially during the beginning of the reaction. Also, its computational time is the shortest one compared to the other methods.

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Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽

10.3390/geosciences9010029 ◽

2019 ◽

Vol 9 (1) ◽

pp. 29 ◽

Cited By ~ 1

Author(s):

Paul M. Delgado ◽

V. M. Krushnarao Kotteda ◽

Vinod Kumar

Keyword(s):

Fixed Point ◽

Asymptotic Solution ◽

Relative Error ◽

Operator Splitting ◽

Splitting Method ◽

Approximate Solutions ◽

Time Step ◽

Operator Splitting Method ◽

The Stability ◽

Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

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Rub Interactions of Flexible Casing Rotor Systems

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.3240308 ◽

1989 ◽

Vol 111 (4) ◽

pp. 652-658 ◽

Cited By ~ 19

Author(s):

F. K. Choy ◽

J. Padovan ◽

C. Batur

Keyword(s):

Numerical Solutions ◽

Spectral Characteristics ◽

Integration Time ◽

Sudden Increase ◽

Time Step ◽

Impeller Blade ◽

Rotor Systems ◽

Machine Failure ◽

The Time Domain ◽

Turbine Impeller

Rub interactions between a rotor assembly and its corresponding casing structure has long been one of the major causes for machine failure. Fracture/fatigue failures of turbine impeller blade components may even lead to catastrophic consequences. This paper presents a comprehensive analysis of a complex rotor-bearing-blade-casing system during component rub interactions. The modal method is used in this study. Orthonormal coupled rotor-casing modes are used to obtain accurate relative motion between rotor and casing. External base vibration input and the sudden increase of imbalance are used to simulate suddenly imposed adversed operating condition. Nonlinear turbine/impeller blade effects are included with the various stages of single/multiple blade participation. A variable integration time step procedure is introduced to insure both accuracy and efficiency in numerical solutions. The dynamic characteristics of the system are examined in both the time domain and the frequency domain using a numerical FFT procedure. Nonlinear bearing and seal forces are also included to enhance a better simulation of the operating system. Frequency components of the system spectral characteristics will be correlated with the localized rub excitations to enable rub signature analysis. A multibearing flexible casing rotor system will be used as an example. Conclusions will be drawn from the results of an extensive parametric study.

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