Assessment of corrected time-step method for nominal ionospheric gradient calculation: A comparative analysis with spatial approaches

The effect of ionospheric delay on the ground-based augmentation system under normal conditions can be mitigated by determining the value of the nominal ionospheric gradient(σvig).The nominal ionospheric gradient is generally obtained from Continuously Operating Reference Stations data by using the spatial single-difference method(mixed-pair,station-pair,or satellite-pair)or the temporal single-difference method(time-step).The time-step method uses only a single receiver,but it still contains ionospheric temporal variations.We introduce a corrected time-step method using a fixed-ionospheric pierce point from the geostationary equatorial orbit satellite and test it through simulations based on the global ionospheric model.We also investigate the effect of satellite paths on the corrected time-step method in the region of the equator,which tends to be in a more north–south direction and to have less coverage for the east–west ionospheric gradient.This study also addresses the limitations of temporal variation correction coverage and recommends using only the correction from self-observations.All processes are developed under simulations because observational data are still difficult to obtain.Our findings demonstrate that the corrected time-step method yieldsσvig values consistent with other approaches.

Time-stepping finite element analysis on the influence of skewed rotors and different skew angles on the losses of squirrel cage asynchronous motors

To study the influence of skewed rotors and different skew angles on the losses of squirrel cage asynchronous motors,a 5.5-kW motor was taken as an example and the multi-sliced field-circuit coupled time stepping finite element method(T-S FEM)was used to analyze the axially non-uniform fundamental and harmonic field distribution characteristics at typical locations in the stator and rotor cores.The major conclusions are:firstly the skewed rotor exhibits a decrease in the harmonic copper losses caused by slot harmonic currents in the stator winding and rotor bars.Secondly,the skewed rotor shifts the non-uniform distribution of field in the axial direction,which leads to more severe saturation and an increase in iron losses.The heavier the load,the more pronounced the increase in iron losses.Furthermore,the influences of different skew angles on motor losses are studied systematically,with skew angles from 0.5 to 1.5 stator tooth pitch.It is found that the lowest total loss occurs at 0.8 stator tooth pitch,and the slot harmonics can be decreased effectively.

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become useful.The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations.The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time.The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order in both space and time.By using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against time.The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

A Unified FastMemory-Saving Time-SteppingMethod for Fractional Operators and Its Applications

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time,especially for long-time integration,which taxes computational resources heavily for high-dimensional problems.Here,we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constant-order fractional operators,and identify the current pitfalls of such methods.In order to overcome the pitfalls,an improved sum-of-exponentials is developed and verified.We also present several sumof-exponentials for the approximation of the kernel function in variable-order fractional operators.Subsequently,based on the sum-of-exponentials,we propose a unified framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable orders.We test the fast method based on several benchmark problems,including fractional initial value problems,the time-fractional Allen-Cahn equation in two and three spatial dimensions,and the Schr¨odinger equation with nonreflecting boundary conditions,demonstrating the efficiency and robustness of the proposed method.The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.

On a Large Time-SteppingMethod for the Swift-Hohenberg Equation

The main purpose of this work is to contrast and analyze a large timestepping numerical method for the Swift-Hohenberg(SH)equation.This model requires very large time simulation to reach steady state,so developing a large time step algorithm becomes necessary to improve the computational efficiency.In this paper,a semi-implicit Euler schemes in time is adopted.An extra artificial term is added to the discretized system in order to preserve the energy stability unconditionally.The stability property is proved rigorously based on an energy approach.Numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches by comparing with the classical scheme.

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.

A varying time-step explicit numerical inte-gration algorithm for solving motion equa-tion

If a traditional explicit numerical integration algorithm is used to solve motion equation in the finite element simulation of wave motion, the time-step used by numerical integration is the smallest time-step restricted by the stability criterion in computational region. However, the excessively small time-step is usually unnecessary for a large portion of computational region. In this paper, a varying time-step explicit numerical integration algorithm is introduced, and its basic idea is to use different time-step restricted by the stability criterion in different computational region. Finally, the feasibility of the algorithm and its effect on calculating precision are verified by numerical test.

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

Investigating the Performance of Twin Marine Propellers in Different Ship Wake Fields Using an Unsteady Viscous and Inviscid Solver

In this study,the performance of a twin-screw propeller under the influence of the wake field of a fully appended ship was investigated using a coupled Reynolds-averaged Navier–Stokes(RANS)/boundary element method(BEM)code.The unsteady BEM is an efficient approach to predicting propeller performance.By applying the time-stepping method in the BEM solver,the trailing vortex sheet pattern of the propeller can be accurately captured at each time step.This is the main innovation of the coupled strategy.Furthermore,to ascertain the effect of the wake field of the ship with acceptable accuracy,a RANS solver was developed.A finite volume method was used to discretize the Navier–Stokes equations on fully unstructured grids.To simulate ship motions,the volume of the fluid method was applied to the RANS solver.The validation of each solver(BEM/RANS)was separately performed,and the results were compared with experimental data.Ultimately,the BEM and RANS solvers were coupled to estimate the performance of a twin-screw propeller,which was affected by the wake field of the fully appended hull.The proposed model was applied to a twin-screw oceanography research vessel.The results demonstrated that the presented model can estimate the thrust coefficient of a propeller with good accuracy as compared to an experimental self-propulsion test.The wake sheet pattern of the propeller in open water(uniform flow)was also compared with the propeller in a real wake field.

Effects of Voltage Sag on the Performance of Induction Motor Based on a New Transient Sequence Component Method

Voltage sag is one of the most common power quality disturbances in industry,which causes huge inrush currents in stator windings of induction motors,and adversely impacts the motor secure operation.This paper firstly introduces a 2D Time-Stepping multi-slice finite element method(2D T-S multi-slice FEM)which is used for calculating the magnetic field distribution in induction motors under different sag events.Then the paper deduces the transient analytical expression of stator inrush current based on the classical theory of AC motors and presents a separation method for the positive,negative and zero sequence values based on instantaneous currents.With this method,the paper studies the influences of voltage sag amplitude,phase-angle jump and initial phase angle on the stator positive-and negative-sequence peak currents of 5.5 kW and 55 kW induction motors.This paper further proposes a motor protection method under voltage sag condition with the stator negative-sequence peak currents as the protection threshold,so that the protection false trip can be avoided effectively.Finally,the calculation and analysis results are validated by the comparison of calculated and measured stator peak value of the 5.5 kW induction motor.

Characteristics of permanent magnet linear synchronous motor fed by spwm inverter based on field-circuit coupled method

Presented field-circuit coupled adaptive time-stepping finite element method to study on permanent magnet linear synchronous motor (PMLSM) characteristics fed by SPWM voltage source inverter.In air-gap field where the direction or magnitude of the field is changing rapidly,the smallest elements are demanded due to high accuracy to use adaptive meshing technique.The co-simulation was used with the status space functions and time-step finite element functions,in which time-step of the status space functions was the smallest than finite element functions'.The magnitude relation of the normal elec- tromagnetic force and tangential electromagnetic force and the period were attained,and current curve was very abrupt at current zero area due to the bigger resistance and leak- age reactance,including main characteristics of motor voltage and velocity.The simulation results compare triumphantly with the experiments results.

Random Timestepping Algorithm with Exponential Distribution for Pricing Various Structures of One-Sided Barrier Options

The exponentially-distributed random timestepping algorithm with boundary test is implemented to evaluate the prices of some variety of single one-sided barrier option contracts within the framework of Black-Scholes model, giving efficient estimation of their hitting times. It is numerically shown that this algorithm, as for the Brownian bridge technique, can improve the rate of weak convergence from order one-half for the standard Monte Carlo to order 1. The exponential timestepping algorithm, however, displays better results, for a given amount of CPU time, than the Brownian bridge technique as the step size becomes larger or the volatility grows up. This is due to the features of the exponential distribution which is more strongly peaked near the origin and has a higher kurtosis compared to the normal distribution, giving more stability of the exponential timestepping algorithm at large time steps and high levels of volatility.

A linear complementarity model for multibody systems with frictional unilateral and bilateral constraints

The Lagrange-I equations and measure differential equations for multibody systems with unilateral and bilateral constraints are constructed. For bilateral constraints, frictional forces and their impulses contain the products of the filled-in relay function induced by Coulomb friction and the absolute values of normal constraint reactions. With the time-stepping impulse-velocity scheme, the measure differential equations are discretized. The equations of horizontal linear complementarity problems （HLCPs）, which are used to compute the impulses, are constructed by decomposing the absolute function and the filled-in relay function. These HLCP equations degenerate into equations of LCPs for frictional unilateral constraints, or HLCPs for frictional bilateral constraints. Finally, a numerical simulation for multibody systems with both unilateral and bilateral constraints is presented.

TIME-DOMAIN RADIATION CHARACTERISTICS OF V-DIPOLE ARRAYS

The time-domain radiation characteristics of a three V-dipole array have been stud-ied by direct time-domain method.Some valuable results are obtained.

Development of a Balanced Adaptive Time‑Stepping Strategy Based on an Implicit JFNK‑DG Compressible Flow Solver

A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iterative errors generated within one time step is constructed.With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization,an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization.The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection,steady-state fow past a fat plate,Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3900.The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady,well-resolved and under-resolved simulations.

Influence of Initial Value of Transient Time Step on Numerical Simulation of Blow Molding Balloon

The correlation between the initial time value of transient iterative parameters and the blowing pressure in the numerical simulation process of blowing balloon is investigated by POLYFLOW. The results show that: 1) As the blow molding pressure increases, the boundary value of the iterative time step decreases rapidly at first and then slowly. At the end of the first step of iterative calculation for each boundary value, the balloon parison is in the mold core cavity. 2) If the initial time value of transient iterative parameters is smaller than the boundary value of the iterative time step, the balloon parison is still in the mold core cavity at the end of the first iteration. However, if the iterative calculation continues, the calculation process may be interrupted when the time step is smaller than the initial time value of the transient iterative parameters, which makes the blow molding simulation of balloon unable to continue. 3) It is suggested that the initial time value of transient iterative parameters is one order of magnitude smaller than the boundary value of the iterative time step to complete smoothly the simulation of blow molding balloon.

ANNOUNCEMENT ON“SHARP ERROR ESTIMATE OF BDF2 SCHEME WITH VARIABLE TIME STEPS FOR LINEAR REACTION-DIFFUSION EQUATIONS”

In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.

Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

Fast High Order and Energy Dissipative Schemes with Variable Time Steps for Time-Fractional Molecular Beam Epitaxial Growth Model

In this paper,we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial(MBE)growth model on the nonuniform mesh.More precisely,(2−α)-order,secondorder and(3−α)-order time-stepping schemes are developed for the timefractional MBE model based on the well known L1,L2-1σ,and L2 formulations in discretization of the time-fractional derivative,which are all proved to be unconditional energy dissipation in the sense of a modified discrete nonlocalenergy on the nonuniform mesh.In order to reduce the computational storage,we apply the sum of exponential technique to approximate the history part of the time-fractional derivative.Moreover,the scalar auxiliary variable(SAV)approach is introduced to deal with the nonlinear potential function and the history part of the fractional derivative.Furthermore,only first order method is used to discretize the introduced SAV equation,which will not affect high order accuracy of the unknown thin film height function by using some proper auxiliary variable functions V(ξ).To our knowledge,it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified L1-,L2-1σ-,and L2-based high-order schemes on the nonuniform mesh,which is essentially important for such time-fractional MBE models with low regular solutions at initial time.Finally,a series of numerical experiments are carried out to verify the accuracy and efficiency of the proposed schemes.