Energy quadratization Runge-Kutta method for the modified phase field crystal equation

Abstract In this paper, we propose high order and unconditionally energy stable methods for a modified phase field crystal equation by applying the strategy of the energy quadratization Runge–Kutta methods. We transform the original model into an equivalent system with auxiliary variables and quadratic free energy. The modified system preserves the laws of mass conservation and energy dissipation with the associated energy functional. We present rigorous proofs of the mass conservation and energy dissipation properties of the proposed numerical methods and present numerical experiments conducted to demonstrate their accuracy and energy stability. Finally, we compare long-term simulations using an indicator function to characterize the pattern formation.

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962316500082 ◽

2016 ◽

Vol 07 (02) ◽

pp. 1650008 ◽

Cited By ~ 4

Author(s):

Beibei Zhu ◽

Zhenxuan Hu ◽

Yifa Tang ◽

Ruili Zhang

Keyword(s):

Numerical Simulation ◽

Hamiltonian System ◽

Second Order ◽

Symplectic Methods ◽

Kutta Method ◽

Numerical Accuracy ◽

Runge Kutta Method ◽

Simulation Results ◽

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system. The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

Energy dissipation–preserving time-dependent auxiliary variable method for the phase-field crystal and the Swift–Hohenberg models

Numerical Algorithms ◽

10.1007/s11075-021-01176-9 ◽

2021 ◽

Author(s):

Junxiang Yang ◽

Junseok Kim

Keyword(s):

Energy Dissipation ◽

Phase Field ◽

Auxiliary Variable ◽

Time Dependent ◽

Variable Method ◽

Numerical approximations for a new L2-gradient flow based Phase field crystal model with precise nonlocal mass conservation

Computer Physics Communications ◽

10.1016/j.cpc.2019.05.006 ◽

2019 ◽

Vol 243 ◽

pp. 51-67 ◽

Cited By ~ 4

Author(s):

Jun Zhang ◽

Xiaofeng Yang

Keyword(s):

Phase Field ◽

Gradient Flow ◽

Mass Conservation ◽

Numerical Approximations ◽

Crystal Model ◽

Numerical Solution of Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Based on Contraharmonic Mean

Indian Journal Of Applied Research ◽

10.15373/2249555x/apr2013/111 ◽

2011 ◽

Vol 3 (4) ◽

pp. 59-63 ◽

Cited By ~ 1

Author(s):

R. Gethsi Sharmila ◽

◽

E. C. Henry Amirtharaj

Keyword(s):

Numerical Solution ◽

Fourth Order ◽

Initial Value Problems ◽

Kutta Method ◽

Initial Value ◽

Runge Kutta Method ◽

Contraharmonic Mean ◽

Numerical Solution of Fuzzy Differential Equations by Seventh Order Runge-Kutta Method

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/1143 ◽

2019 ◽

Vol 10 (7) ◽

pp. 1518-1528

Author(s):

P. Rajkumar ◽

S. Rubanraj

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Kutta Method ◽

Runge Kutta Method ◽

Seventh Order ◽

A Comparative Study on Numerical Solution of Initial Value Problem by Using Euler�s Method, Modified Euler�s Method and Runge-Kutta Method

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/784 ◽

2018 ◽

Vol 9 (5) ◽

pp. 493-500

Author(s):

Md. Kamruzzaman ◽

Mithun Chandra Nath

Keyword(s):

Comparative Study ◽

Numerical Solution ◽

Initial Value Problem ◽

Kutta Method ◽

Initial Value ◽

Runge Kutta Method ◽

China-Ireland International Conference on Information and Communications Technologies (CIICT 2007) ◽

A modified embedded Runge-Kutta method for cloth simulation

10.1049/cp:20070810 ◽

2007 ◽

Author(s):

Xinrong Hu ◽

Lan Wei

Keyword(s):

Kutta Method ◽

Cloth Simulation ◽

Runge Kutta Method ◽

Enabling simulations of grains within a full rotation range in amplitude expansion of the phase-field crystal model

Physical Review E ◽

10.1103/physreve.101.043309 ◽

2020 ◽

Vol 101 (4) ◽

Author(s):

Matjaž Berčič ◽

Goran Kugler

Keyword(s):

Phase Field ◽

Crystal Model ◽

Full Rotation ◽

Thermodensity coupling in phase-field-crystal-type models for the study of rapid crystallization

Physical Review Materials ◽

10.1103/physrevmaterials.3.053804 ◽

2019 ◽

Vol 3 (5) ◽

Cited By ~ 2

Author(s):

Gabriel Kocher ◽

Nikolas Provatas

Keyword(s):

Phase Field ◽

Rapid Crystallization ◽

Crystal Type ◽

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

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0167 ◽

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 ◽

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.