A Numerical Solution to the Point Kinetic Equations Using Diagonally Implicit Runge Kutta Method

2016 ◽  
Author(s):  
Zhizhu Zhang ◽  
Yun Cai
Keyword(s):  
Kinetic Equations ◽  
Fixed Time ◽  
Time Step ◽  
Perfect Agreement ◽  
Computation Efficiency ◽  
Runge Kutta Method ◽  
Adaptive Time Step ◽  
Long Time ◽  
Runge Kutta ◽  
Point Kinetics

It would take a long time to solve the point kinetics equations by using full implicit Runge-Kutta (FIRK) for the strong stiffness. Diagonally implicit Runge-Kutta (DIRK) is a useful tool like FIRK to solve the stiff differential equations, while it could greatly reduce the computation compared to FIRK. By embedded low-order Runge-Kutta, DIRK is implemented with the time step adaptation technique, which improves the computation efficiency of DIRK. Through four typical cases with step, ramp sinusoidal and zig-zag reactivity insertions, it shows that the results obtained by DIRK are in perfect agreement with other available results and DIRK with adaptive time step technique has more efficiency than DIRK with the fixed time step.

Application of the piecewise constant method in nonlinear dynamics of drill string

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jialin Tian ◽  
Jie Wang ◽  
Yi Zhou ◽  
Lin Yang ◽  
Changyue Fan ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Drill String ◽  
Vibration Velocity ◽  
Kutta Method ◽  
Time Step ◽  
Piecewise Constant ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract Aiming at the current development of drilling technology and the deepening of oil and gas exploration, we focus on better studying the nonlinear dynamic characteristics of the drill string under complex working conditions and knowing the real movement of the drill string during drilling. This paper firstly combines the actual situation of the well to establish the dynamic model of the horizontal drill string, and analyzes the dynamic characteristics, giving the expression of the force of each part of the model. Secondly, it introduces the piecewise constant method (simply known as PT method), and gives the solution equation. Then according to the basic parameters, the axial vibration displacement and vibration velocity at the test points are solved by the PT method and the Runge–Kutta method, respectively, and the phase diagram, the Poincare map, and the spectrogram are obtained. The results obtained by the two methods are compared and analyzed. Finally, the relevant experimental tests are carried out. It shows that the results of the dynamic model of the horizontal drill string are basically consistent with the results obtained by the actual test, which verifies the validity of the dynamic model and the correctness of the calculated results. When solving the drill string nonlinear dynamics, the results of the PT method is closer to the theoretical solution than that of the Runge–Kutta method with the same order and time step. And the PT method is better than the Runge–Kutta method with the same order in smoothness and continuity in solving the drill string nonlinear dynamics.

Parallel Computation of the Point Neutron Kinetic Equations Using Parallel Revisionist Integral Deferred Correction

2018 ◽  
Author(s):  
Yun Cai ◽  
Xingjie Peng ◽  
Qing Li ◽  
Zhizhu Zhang ◽  
Zhumin Jiang ◽  
...  
Keyword(s):  
Time Integration ◽  
Kinetic Equations ◽  
Accurate Method ◽  
Small Time ◽  
Time Step ◽  
Deferred Correction ◽  
Time Integration Method ◽  
Speed Up ◽  
Point Kinetics ◽  
Reactivity Insertion

The point kinetics is very important to the safety of the reactor operation. However, these equations are stiff and usually solved with very small time step. These equations are solved by Revisionist integral deferred correction (RIDC), which is a parallel time integration method. RIDC is a highly accurate method, and it reduces the error by iteration. Based on C++ and MPI, a four-core fourth-order RIDC is implemented and tested by several cases, such as step, ramp, and sinusoidal reactivity insertion. Compared with other methods, the time step of RIDC in the step reactivity insertion case is smaller, but it’s larger in the case of the sinusoidal reactivity insertion. RIDC can keep high accuracy while the time step is appropriately large. The numerical results also show that the speed-up ratio can achieve 2 when 4 processors are used.

Long-time symplectic integration: the example of four-vortex motion

1991 ◽  
Vol 432 (1886) ◽  
pp. 481-494 ◽  
Keyword(s):  
Fourth Order ◽  
Initial Conditions ◽  
Hamiltonian Formulation ◽  
Superior Performance ◽  
Integration Time ◽  
Characteristic Time Scale ◽  
Symplectic Integration ◽  
Time Step ◽  
Long Time ◽  
Runge Kutta

Detailed comparisons are made between long-time numerical integration of the motion of four identical point vortices obtained using both a fourth-order symplectic integration method of the implicit Runge-Kutta type and a standard fourth-order explicit Runge-Kutta scheme. We utilize the reduced hamiltonian formulation of the four-vortex problem due to Aref & Pomphrey. Initial conditions which give both fully chaotic and also quasi-periodic motions are considered over integration times of order 10 6 -10 7 times the characteristic time scale of the evolution. The convergence, as the integration time step is decreased, of the Poincaré section is investigated. When smoothness of the section compared to the converged image, and the fractional change in the hamiltonian H are used as diagnostic indicators, it is found that the symplectic scheme gives substantially superior performance over the explicit scheme, and exhibits only an apparent qualitative degrading in results up to integration time steps of order the minimum timescale of the evolution. It is concluded that this performance derives from the symplectic rather than the implicit character of the method.

scholarly journals Numerical calculation for coupling vibration system by Piecewise-Laplace method

2020 ◽  
Vol 103 (3) ◽  
pp. 003685042093855
Author(s):  
Pan Fang ◽  
Kexin Wang ◽  
Liming Dai ◽  
Chixiang Zhang
Keyword(s):  
Numerical Computation ◽  
Laplace Transformation ◽  
Original System ◽  
Continuity Condition ◽  
Kutta Method ◽  
Laplace Method ◽  
Time Step ◽  
Coupling Vibration ◽  
Runge Kutta Method ◽  
Runge Kutta

To improve the reliability and accuracy of dynamic machine in design process, high precision and efficiency of numerical computation is essential means to identify dynamic characteristics of mechanical system. In this paper, a new computation approach is introduced to improve accuracy and efficiency of computation for coupling vibrating system. The proposed method is a combination of piecewise constant method and Laplace transformation, which is simply called as Piecewise-Laplace method. In the solving process of the proposed method, the dynamic system is first sliced by a series of continuous segments to reserve physical attribute of the original system; Laplace transformation is employed to separate coupling variables in segment system, and solutions of system in complex domain can be determined; then, considering reverse Laplace transformation and residues theorem, solution in time domain can be obtained; finally, semi-analytical solution of system is given based on continuity condition. Through comparison of numerical computation, it can be found that precision and efficiency of numerical results with the Piecewise-Laplace method is better than Runge-Kutta method within same time step. If a high-accuracy solution is required, the Piecewise-Laplace method is more suitable than Runge-Kutta method.

scholarly journals Acceleration of Runge-Kutta integration schemes

2004 ◽  
Vol 2004 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Phailaung Phohomsiri ◽  
Firdaus E. Udwadia
Keyword(s):  
Computational Cost ◽  
Fixed Time ◽  
Lower Number ◽  
Integration Scheme ◽  
Time Step ◽  
Third Order ◽  
Numerical Examples ◽  
Integration Schemes ◽  
Local Accuracy ◽  
Runge Kutta

A simple accelerated third-order Runge-Kutta-type, fixed time step, integration scheme that uses just two function evaluations per step is developed. Because of the lower number of function evaluations, the scheme proposed herein has a lower computational cost than the standard third-order Runge-Kutta scheme while maintaining the same order of local accuracy. Numerical examples illustrating the computational efficiency and accuracy are presented and the actual speedup when the accelerated algorithm is implemented is also provided.

scholarly journals Trajectory errors of different numerical integration schemes diagnosed with the MPTRAC advection module driven by ECMWF operational analyses

2018 ◽  
Vol 11 (2) ◽  
pp. 575-592 ◽  
Author(s):  
Thomas Rößler ◽  
Olaf Stein ◽  
Yi Heng ◽  
Paul Baumeister ◽  
Lars Hoffmann
Keyword(s):  
Integration Scheme ◽  
Kutta Method ◽  
Time Step ◽  
Third Order ◽  
Truncation Errors ◽  
The Third ◽  
Trajectory Calculations ◽  
Integration Schemes ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract. The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge–Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge–Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge–Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge–Kutta method, respectively.

A Modified Implicit Iterative Difference Algorithm for Stiff Chemical Kinetic Equations in Complex Combustion System

2011 ◽  
Vol 295-297 ◽  
pp. 2333-2340 ◽  
Author(s):  
Bo He ◽  
Wan Sheng Nie ◽  
Song Jiang Feng ◽  
Guo Qiang Li
Keyword(s):  
Mass Fraction ◽  
Kinetic Equations ◽  
Operator Method ◽  
Chemical Kinetic ◽  
Chemical Mechanism ◽  
Combustion System ◽  
Time Step ◽  
Nitrogen Tetroxide ◽  
Elementary Reactions ◽  
Computation Efficiency

The splitting-operator method has been widely used in the numerical simulation of complex combustion system with its high computation efficiency. The main difficult encountered in the method is how to integrate the stiff chemical kinetics ordinary differential equation (ODE) efficiently and accurately in each grid node of flow field. Although several ODE software packages such as LSODE and VODE have very predominant performance in integrating stiff and non-stiff case, it couldn’t ensure the positivity of mass fraction in integrating stiff chemical kinetic ODE. A detail liquid rocket hypergolic bipropellant combination of Monomethylhydrazine (MMH)/Red Fuming Nitric Acid (RFNA) chemical mechanism consisting of 550 elementary reactions among 83 species was constituted basing on several relevant literatures firstly. And then a modified implicit iterative difference algorithm with variable time steps was developed to tackle the problem of mass fraction non-positive. The results from the integration of MMH /nitrogen tetroxide (NTO) chemical kinetic ODE indicate that the algorithm is always retain stability and positive value of mass fraction. The time step length would also be automatically adjusted at different chemical reaction time in the algorithm to keep its computational efficiency.

scholarly journals Preliminary results of the ion trajectory tracking in the acceleration region of the VINCY Cyclotron

2006 ◽  
Vol 16 (1) ◽  
pp. 5-8
Author(s):  
Andjelija Ilic ◽  
Jasna Ristic-Djurovic ◽  
Sasa Cirkovic
Keyword(s):  
Trajectory Tracking ◽  
Computer Code ◽  
Acceleration Region ◽  
Time Step ◽  
Ion Trajectory ◽  
Accuracy Requirement ◽  
Runge Kutta Method ◽  
Orbital Frequency ◽  
Adaptive Time Step ◽  
Adaptive Time

In an accelerating region of a cyclotron an ion makes a large number of turns; thus its tracking requires fast as well as highly accurate computation. Computer code, based on the adaptive time step fourth order Runge Kutta method, has been developed. Accuracy requirement is set simultaneously on the position and momentum calculation. Magnetic fields used as input, have been evaluated in terms of the radial fluctuations of the orbital frequency, i.e. their isochronism. Ion trajectory tracking has been performed for the four test beams: H-, H2 +, 4He+, and 40Ar6+.

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