By Xinyuan Wu, Kai Liu, Wei Shi
This publication describes quite a few powerful and effective structure-preserving algorithms for second-order oscillatory differential equations. Such platforms come up in lots of branches of technological know-how and engineering, and the examples within the booklet contain platforms from quantum physics, celestial mechanics and electronics. To competently simulate the real habit of such structures, a numerical set of rules needs to defend up to attainable their key structural houses: time-reversibility, oscillation, symplecticity, and effort and momentum conservation. The e-book describes novel advances in RKN equipment, ERKN tools, Filon-type asymptotic equipment, AVF equipment, and trigonometric Fourier collocation equipment. The accuracy and potency of every of those algorithms are established through cautious numerical simulations, and their structure-preserving homes are conscientiously validated by means of theoretical research. The e-book additionally offers insights into the sensible implementation of the methods.
This e-book is meant for engineers and scientists investigating oscillatory platforms, in addition to for academics and scholars who're drawn to structure-preserving algorithms for differential equations.
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Additional resources for Structure-Preserving Algorithms for Oscillatory Differential Equations II
The results are shown in Fig. 8. In Fig. 01, hence the point is not plotted in the graph. 6 log 10 (Function evaluations) Fig. 5 with different ω. 6 ) Fig. 5 with different ω. 001 on the intervals [0, 2i × 10], i = 2, . . , 5. The results for different ω = 100 and 200 are shown in Fig. 9. For this Hamiltonian problem, long time-step methods may lead to the problem of numerically induced resonance instabilities. 02). We consider the long-time interval [0, 1000]. 11 present the results of the different methods.
They are expected to have better numerical behaviour than the classical Störmer–Verlet formula. The key point here is that each new multi-frequency and multidimensional Störmer–Verlet formula utilizes a combination of existing trigonometric integrators and symplectic schemes. 1) are presented below. 1 Improved Störmer–Verlet Formula 1 The first improved Störmer–Verlet formula is based on the multi-frequency and multidimensional ARKN schemes and the corresponding symplectic conditions. 1) (see ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s Yi = yn + ci hyn + h 2 a¯ i j f (tn + c j h, Y j ) − MY j , i = 1, 2, .
Denote f 1 = f (Y1 ). 9) yn+1 = φ0 (h 2 m J J )ynJ + φ1 (h 2 m J J )hynJ + h 2 b¯1 (h 2 m J J ) f 1J , ⎪ ⎩ J 2 J 2 J 2 J yn+1 = −hm J J φ1 (h m J J )yn + φ0 (h m J J )yn + hb1 (h m J J ) f 1 , where the superscript J (J = 1, 2, . . , d) denotes the J th component of a vector. 6) is identical to d J =1 J J dyn+1 ∧ dyn+1 = d J =1 dynJ ∧ dynJ . To show this equality, it is required to compute J J dyn+1 ∧ dyn+1 = [φ02 (h 2 m J J ) + h 2 m J J φ12 (h 2 m J J )]dynJ ∧ dynJ + h[b1 (h 2 m J J )φ0 (h 2 m J J ) + b¯1 (h 2 m J J )h 2 m J J φ1 (h 2 m J J )]dynJ ∧ d f 1J + h 2 [b1 (h 2 m J J )φ1 (h 2 m J J ) − b¯1 (h 2 m J J )φ0 (h 2 m J J )]dynJ ∧ d f 1J .