scholarly journals Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1158
Author(s):  
Yonghyeon Jeon ◽  
Soyoon Bak ◽  
Sunyoung Bu
Keyword(s):  
Stiff Systems ◽  
Stiff Problem ◽  
Initial Value ◽  
Step Method ◽  
Stiff System ◽  
Computational Costs ◽  
Numerical Tests ◽  
Multi Stage ◽  
Enormous Amount ◽  
The Difference

In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a massive linear system provided by the linearization of a highly stiff system. We investigate the possibility of usage of multi-stage methods for stiff systems by discussing the difference between the two methods in several numerical experiments. Moreover, the advantages of multi-stage methods are heuristically presented even for nonlinear stiff systems through several numerical tests.

EXPLICIT EIGHTH ORDER TWO-STEP METHODS WITH NINE STAGES FOR INTEGRATING OSCILLATORY PROBLEMS

2006 ◽  
Vol 17 (06) ◽  
pp. 861-876 ◽  
Author(s):  
Ch. TSITOURAS
Keyword(s):  
Initial Value Problem ◽  
Second Order ◽  
Initial Value ◽  
Step Method ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
Phase Errors ◽  
Matlab Program ◽  
Algebraic Order

We present a new explicit hybrid two step method for the solution of second order initial value problem. It costs only nine function evaluations per step and attains eighth algebraic order so it is the cheapest in the literature. Its coefficients are chosen to reduce amplification and phase errors. Thus the method is well suited for facing problems with oscillatory solutions. After implementing a MATLAB program, we proceed with numerical tests that justify our effort.

scholarly journals Seventh order hybrid block method for solution of first order stiff systems of initial value problems

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
O. A. Akinfenwa ◽  
R. I. Abdulganiy ◽  
B. I. Akinnukawe ◽  
S. A. Okunuga
Keyword(s):  
Initial Value Problems ◽  
Stiff Systems ◽  
Block Method ◽  
Initial Value ◽  
First Order ◽  
Seventh Order

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

A difference scheme for a stiff system of conservation laws

1998 ◽  
Vol 128 (6) ◽  
pp. 1403-1414 ◽  
Author(s):  
Wen-An Yong
Keyword(s):  
Relaxation Time ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Difference Method ◽  
Initial Value Problems ◽  
Equilibrium System ◽  
Initial Value ◽  
Stiff System ◽  
System Of Conservation Laws ◽  
Entropy Satisfying

A stiff system of conservation laws is analysed using a difference method. The existence of entropy-satisfying BV-solutions to the initial value problems is established. Furthermore, we show that the solutions converge to the solutions of the corresponding equilibrium system as the relaxation time tends to zero.

scholarly journals Energy Expenditure of a Single Sit-to-Stand Movement with Slow Versus Normal Speed Using the Different Frequency Accumulation Method

Medicina ◽  
2019 ◽  
Vol 55 (3) ◽  
pp. 77 ◽  
Author(s):  
Takashi Nakagata ◽  
Yosuke Yamada ◽  
Yoichi Hatamoto ◽  
Hisashi Naito
Keyword(s):  
Random Order ◽  
Young Males ◽  
Energy Expenditures ◽  
Metabolic Equivalents ◽  
Sit To Stand ◽  
Multi Stage ◽  
Study Results ◽  
Normal Speed ◽  
The Difference ◽  
Accumulation Method

Background and objectives: The purpose of this study was to compare the energy expenditures (EE) of a single sit-to-stand (STS) movements with slow and normal speeds using a multi-stage exercise test. Materials and Methods: Twelve young males, aged 21–27 years (age, 23.0 ± 1.7 years; height, 171.2 ± 6.1 cm; weight, 64.3 ± 5.6 kg), performed repeated 3-s stand-up and 3-s sit-down (slow) or 1-s stand-up and 1-s sit-down (normal) movement on two different days with random order. All the participants completed multi-stage tests at different STS frequencies per minute. The slope and intercept of the linear regression relationship between the EE (kcal/min) and the STS frequency were obtained, and the slope of the regression was quantified as the EE of an STS. Results: The metabolic equivalents (METs) of the STS-slow was 4.5 METs for the frequency of 10 times/min (in total 1 min), and the net EE was 5.00 ± 1.2 kcal/min. The net EE of the STS-slow was 0.37 ± 0.12 kcal, which was significantly greater than that during the STS-normal (0.26 ± 0.06 kcal). The difference between the EEs of the STS-slow and STS-normal was significantly greater in taller and heavier subjects. Conclusions: We concluded that the intensity of STS-slow movement is moderate, and the EE during an STS-slow (0.37 ± 0.12 kcal) is higher than that during an STS-normal (0.26 ± 0.06 kcal). Our study results will help exercise and/or health professionals prescribe physical activity programs using STS movement for healthy young population groups.

A nonlinear Wiener degradation model integrating degradation data under accelerated stresses and real operating environment

2020 ◽  
pp. 1748006X2097809
Author(s):  
Li Sun ◽  
Fangchao Zhao ◽  
Narayanaswamy Balakrishnan ◽  
Honggen Zhou ◽  
Xiaohui Gu
Keyword(s):  
Remaining Useful Life ◽  
Normal Stresses ◽  
Operating Environment ◽  
Unknown Parameters ◽  
Step Method ◽  
Degradation Model ◽  
Accelerated Degradation ◽  
Degradation Data ◽  
Proposed Model ◽  
The Difference

Remaining useful life (RUL) prediction in real operating environment (ROE) plays an important role in condition-based maintenance. However, the life information in ROE is limited, especially for some long-life products. In such cases, accelerated degradation test (ADT) is an effective method to collect data and then the accelerated degradation data are converted to normal level of accelerated stresses through acceleration factors. However, the stresses in ROE are different from normal stresses since there are some other stresses except normal stresses, which cannot be accelerated, but still have impact on the degradation. To predict the RUL in ROE, a nonlinear Wiener degradation model is proposed based on failure mechanism invariant principle which is the precondition and requirement of an ADT and a calibration factor is introduced to calibrate the difference between ROE and normal stresses. Moreover, the unit-to-unit variability is considered in the concern model. Based upon the proposed approach, the RUL distribution is derived in closed form. The unknown parameters in the model are obtained by a new two-step method through fuzing converted degradation data in normal stresses and degradation data in ROE. Finally, the validity of the proposed model is demonstrated through several simulation data and a case study.

scholarly journals A sixth-order A-stable explicit one-step method for stiff systems

1998 ◽  
Vol 35 (9) ◽  
pp. 59-64 ◽  
Author(s):  
Xin-Yuan Wu
Keyword(s):  
Sixth Order ◽  
Stiff Systems ◽  
Step Method ◽  
One Step

A P-STABLE EIGHTEENTH-ORDER SIX-STEP METHOD FOR PERIODIC INITIAL VALUE PROBLEMS

2007 ◽  
Vol 18 (03) ◽  
pp. 419-431 ◽  
Author(s):  
CHUNFENG WANG ◽  
ZHONGCHENG WANG
Keyword(s):  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Initial Value ◽  
Step Method ◽  
Periodic Initial Value Problems

In this paper we present a new kind of P-stable eighteenth-order six-step method for periodic initial-value problems. We add the fourth derivatives to our previous P-stable six-step method to increase the accuracy. We apply two classes of well-known problems to our new method and compare it with the previous methods. The numerical results show that the new method is much more stable, accurate and efficient than the previous methods.

STUDY OF A GROWTH ALGORITHM FOR A FEEDFORWARD NETWORK

1989 ◽  
Vol 01 (01) ◽  
pp. 55-59 ◽  
Author(s):  
Jean-Pierre Nadal
Keyword(s):  
Feedforward Network ◽  
Numerical Tests ◽  
The Difference

We study an algorithm for a feedforward network which is similar in spirit to the Tiling algorithm recently introduced: the hidden units are added one by one until the network performs the desired task, and convergence is guaranteed. The difference is in the architecture of the network, which is more constrained here. Numerical tests show performances similar to that of the Tiling algorithm, although the total number of couplings in general grows faster.

scholarly journals Block Backward Differentiation Formulas for Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
T. A. Biala ◽  
S. N. Jator
Keyword(s):  
Numerical Approximation ◽  
Initial Value ◽  
One Dimensional ◽  
Numerical Tests ◽  
Numerical Integrators ◽  
Backward Differentiation Formulas ◽  
Collocation Approach ◽  
Fractional Differential ◽  
Large Systems ◽  
Differentiation Formulas

This paper concerns the numerical approximation of Fractional Initial Value Problems (FIVPs). This is achieved by constructing k-step continuous BDFs. These continuous schemes are developed via the interpolation and collocation approach and are used to obtain the discrete k-step BDF and (k-1) additional methods which are applied as numerical integrators in a block-by-block mode for the integration of FIVP. The properties of the methods are established and regions of absolute stability of the methods are plotted in the complex plane. Numerical tests including large systems arising form the semidiscretization of one-dimensional fractional Burger’s equation show that the methods are highly accurate and efficient.

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