ERKN integrators solving multi-frequency highly oscillatory systems with applications

2018 ◽  
Author(s):  
Xinyuan Wu ◽  
Bin Wang
Keyword(s):  
Oscillatory Systems ◽  
Highly Oscillatory ◽  
Erkn Integrators ◽  
Highly Oscillatory Systems

scholarly journals Double-Loop Control Structure for Rotary Drum Granulation Loop

Processes ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1423
Author(s):  
Ludmila Vesjolaja ◽  
Bjørn Glemmestad ◽  
Bernt Lie
Keyword(s):  
Control Structure ◽  
Control Strategies ◽  
Double Loop ◽  
The Novel ◽  
Loop Control ◽  
Process Dynamics ◽  
Oscillatory Systems ◽  
Novel Approach ◽  
Highly Oscillatory ◽  
Highly Oscillatory Systems

The operation of granulation plants on an industrial scale is challenging. Periodic instability associated with the operation of the granulation loop causes the particle size distribution of the particles flowing out from the granulator to oscillate, thus making it difficult to maintain the desired product quality. To address this problem, two control strategies are proposed in this paper, including a novel approach, where product-sized particles are recycled back to maintain a stable granulation loop process. A dynamic model of the process that is based on a population balance equation is used to represent the process dynamics. Both of the control strategies utilize a double-loop control structure that is suitable for highly oscillatory systems. The simulation results show that both control strategies, including the novel approach, are able to remove the oscillating behaviour and stabilize the granulation plant loop.

An Essential Extension of the Finite-Energy Condition for Extended Runge-Kutta-Nyström Integrators when Applied to Nonlinear Wave Equations

2017 ◽  
Vol 22 (3) ◽  
pp. 742-764 ◽  
Author(s):  
Lijie Mei ◽  
Changying Liu ◽  
Xinyuan Wu
Keyword(s):  
Error Analysis ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Energy Condition ◽  
Finite Energy ◽  
Nonlinear Wave Equations ◽  
Oscillatory Systems ◽  
Highly Oscillatory ◽  
Runge Kutta ◽  
Erkn Integrators

AbstractThis paper is devoted to an extension of the finite-energy condition for extended Runge-Kutta-Nyström (ERKN) integrators and applications to nonlinear wave equations. We begin with an error analysis for the integrators for multi-frequency highly oscillatory systems , where M is positive semi-definite, . The highly oscillatory system is due to the semi-discretisation of conservative, or dissipative, nonlinear wave equations. The structure of such a matrix M and initial conditions are based on particular spatial discretisations. Similarly to the error analysis for Gaustchi-type methods of order two, where a finite-energy condition bounding amplitudes of high oscillations is satisfied by the solution, a finite-energy condition for the semi-discretisation of nonlinear wave equations is introduced and analysed. These ensure that the error bound of ERKN methods is independent of . Since stepsizes are not restricted by frequencies of M, large stepsizes can be employed by our ERKN integrators of arbitrary high order. Numerical experiments provided in this paper have demonstrated that our results are truly promising, and consistent with our analysis and prediction.

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