scholarly journals Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 780
Author(s):  
Aleksandra Tutueva ◽  
Timur Karimov ◽  
Denis Butusov
Keyword(s):  
Simulation Software ◽  
Van Der Pol Oscillator ◽  
Great Success ◽  
General Description ◽  
Implicit Integration ◽  
Variable Stepsize ◽  
Integration Methods ◽  
Computational Costs ◽  
Accuracy Order ◽  
Predictor Corrector

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

scholarly journals Paracoherent Answer Set Semantics meets Argumentation Frameworks

2019 ◽  
Vol 19 (5-6) ◽  
pp. 688-704
Author(s):  
GIOVANNI AMENDOLA ◽  
FRANCESCO RICCA
Keyword(s):  
Logic Programming ◽  
Logic Program ◽  
Great Success ◽  
Argumentation Framework ◽  
Computational Costs ◽  
Abstract Argumentation ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

AbstractIn the last years, abstract argumentation has met with great success in AI, since it has served to capture several non-monotonic logics for AI. Relations between argumentation framework (AF) semantics and logic programming ones are investigating more and more. In particular, great attention has been given to the well-known stable extensions of an AF, that are closely related to the answer sets of a logic program. However, if a framework admits a small incoherent part, no stable extension can be provided. To overcome this shortcoming, two semantics generalizing stable extensions have been studied, namely semi-stable and stage. In this paper, we show that another perspective is possible on incoherent AFs, called paracoherent extensions, as they have a counterpart in paracoherent answer set semantics. We compare this perspective with semi-stable and stage semantics, by showing that computational costs remain unchanged, and moreover an interesting symmetric behaviour is maintained.

A high-order predictor-corrector method for solving nonlinear differential equations of fractional order

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Thien Binh Nguyen ◽  
Bongsoo Jang
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Nonlinear Differential Equations ◽  
Prediction Method ◽  
High Order ◽  
New Class ◽  
Accuracy Order ◽  
Predictor Corrector ◽  
Uniform Accuracy

AbstractAn accurate and efficient new class of predictor-corrector schemes are proposed for solving nonlinear differential equations of fractional order. By introducing a new prediction method which is explicit and of the same accuracy order as that of the correction stage, the new schemes achieve a uniform accuracy order regardless of the values of fractional order

scholarly journals Simulation of heat transfer with considering permafrost thawing in 3D media

2021 ◽  
Vol 2099 (1) ◽  
pp. 012007
Author(s):  
D A Karavaev ◽  
Y M Laevsky
Keyword(s):  
Heat Transfer ◽  
Parallel Algorithm ◽  
High Performance ◽  
Splitting Method ◽  
Computing System ◽  
Phase Changes ◽  
Iteration Step ◽  
Accuracy Order ◽  
Predictor Corrector ◽  
Permafrost Thawing

Abstract An approach to mathematical modeling of heat transfer with a permafrost algorithm in 3D media based on the idea of localizing the phase transition area is considered. The paper presents a problem statement for a non-stationary heat transfer and a description of a numerical method based on a predictor-corrector scheme. For a better understanding of the proposed splitting method, the accuracy order of approximation considering inhomogeneous right-hand side was studied. The phase changes in the numerical implementation of permafrost thawing is considered in the temperature range and requires recalculation of coefficients values of the heat equation at each iteration step with respect to time. A brief description of the parallel algorithm based on a 3D decomposition method and the parallel sweep method is presented. A study of the parallel algorithm implementations using a high-performance computing system of the Siberian Supercomputer Center of the SB RAS was performed. The results of the permafrost algorithm on models with wellbores are also presented.

Asymptotic Integration Methods Applied to Rotating Beams

1980 ◽  
Vol 47 (4) ◽  
pp. 884-890 ◽  
Author(s):  
C. R. Steele ◽  
K. E. Barry
Keyword(s):  
General Theory ◽  
Equations Of Motion ◽  
Vector Form ◽  
Rotating Beams ◽  
Integration Methods ◽  
Computational Costs ◽  
Tensile Forces ◽  
Vibrational Characteristics ◽  
Steady Rotation ◽  
Analytical Complexity

The in-plane vibrational characteristics of an off-axis clamped beam subjected to either compressive or tensile forces arising from steady rotation are studied. The differential equations of motion are cast into state vector form and solved using asymptotic matrix integration methods. The general theory of these methods is described in this paper and their application to the analysis of rotating beams is made. The advantages inherent in these methods with regard to accuracy, reduction of analytical complexity, and savings in computational costs are discussed.

Implicit Integration in Molecular Dynamics Simulation

2008 ◽  
Author(s):  
Nicholas P. Schafer ◽  
Radu Serban ◽  
Dan Negrut
Keyword(s):  
Molecular Dynamics ◽  
Newton Method ◽  
Dynamics Simulation ◽  
Integration Scheme ◽  
Implicit Integration ◽  
Step Size ◽  
Time Step ◽  
Integration Methods ◽  
Size Limitation ◽  
Quasi Newton

Molecular Dynamics (MD) simulation is a versatile methodology that has found many applications in materials science, chemistry and biology. In biology, the models employed range from mixed quantum mechanical and fully atomistic to united atom and continuum mechanical. These systems are evolved in discrete time by solving Newton’s equations of motion at each time step. The numerical methods currently in use limit the step size of a typical all atom simulation to 1 femtosecond. This step size limitation means that many steps need to be taken in order to reach biologically relevant time scales. At each time step, an evaluation of the forces on each atom must be performed resulting in heavy computational loads. This work investigates the use of implicit integration methods in MD. Implicit integration methods have been proven superior to their explicit counterparts in classical mechanical simulation, with which MD has many similarities. Longer time steps reduce the number of force evaluations that must be performed and the corresponding computational load. Herein we present results that compare implicit integration techniques with the current standard for molecular dynamics, the explicit velocity Verlet integration scheme. Total energy conservation is used as a metric for evaluating the dependability of simulations in the microcanonical ensemble. In order to understand the nature of the problem, several long simulations were run and analyzed by performing a Fourier analysis on the position, velocity and acceleration signals. Lastly, several methods for improving the viability of implicit integration methods are considered including replacing the Jacobian used in the Quasi-Newton method with a constant, diagonal mass matrix, evaluating the Jacobian infrequently and finding a better prediction of the system configuration to improve the convergence of the Quasi-Newton method.

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