Steepest descent reaction path integration using a first-order predictor–corrector method

2010 ◽  
Vol 133 (22) ◽  
pp. 224101 ◽  
Author(s):  
Hrant P. Hratchian ◽  
Michael J. Frisch ◽  
H. Bernhard Schlegel
Keyword(s):  
Path Integration ◽  
Reaction Path ◽  
Steepest Descent ◽  
First Order ◽  
Predictor Corrector

Continuous linear multistep method for the general solution of first order initial value problems for Volterra integro-differential equations

2018 ◽  
Vol 14 (5) ◽  
pp. 960-969
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Terhemen Aboiyar
Keyword(s):  
General Solution ◽  
Approximation Method ◽  
Legendre Polynomial ◽  
Basis Function ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Content Type ◽  
First Order ◽  
Predictor Corrector

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

Parallel iteration of two-step Runge-Kutta methods

2021 ◽  
Vol 66 (1) ◽  
pp. 12-24
Author(s):  
Thuy Nguyen Thu
Keyword(s):  
Iteration Process ◽  
Parallel Computers ◽  
Initial Value ◽  
Parallel Methods ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Parallel Iteration ◽  
Predictor Corrector ◽  
First Order Differential Equations

In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

scholarly journals A Comparative Study of First-Order Reliability Method-Based Steepest Descent Search Directions for Reliability Analysis of Steel Structures

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Hamed Makhduomi ◽  
Behrooz Keshtegar ◽  
Mehdi Shahraki
Keyword(s):  
Reliability Index ◽  
Steel Structures ◽  
Steepest Descent ◽  
Building Code ◽  
Step Size ◽  
First Order Reliability Method ◽  
First Order ◽  
Confidence Levels ◽  
Reliability Method ◽  
Bending Capacity

Three algorithms of first-order reliability method (FORM) using steepest descent search direction are compared to evaluate the reliability index of structural steel problems which are designed by the Iranian National Building code. The FORM formula is modified based on a dynamic step size which is computed based on the merit functions named modified Hasofer-Lind and Rackwitz-Fiessler (MHL-RF) method. The efficiency of the gradient, HL-RF, and MHL-RF method was compared for a bar structure under tensile capacity, a multispan beam under bending capacity, a connection under tension load, and a column under axial force. The results illustrated that the MHL-RF method is more efficient than the HL-RF and gradient method. The designed steel components by the Iranian National Building code showed good confidence levels with the reliability index in the range from 2.5 to 3.0.

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