By Xinyuan Wu, Xiong You, Bin Wang
Structure-Preserving Algorithms for Oscillatory Differential Equations describes various powerful and effective structure-preserving algorithms for second-order oscillatory differential equations through the use of theoretical research and numerical validation. Structure-preserving algorithms for differential equations, in particular for oscillatory differential equations, play a tremendous position within the exact simulation of oscillatory difficulties in technologies and engineering. The ebook discusses novel advances within the ARKN, ERKN, two-step ERKN, Falkner-type and energy-preserving tools, and so on. for oscillatory differential equations. The paintings is meant for scientists, engineers, academics and scholars who're attracted to structure-preserving algorithms for differential equations. Xinyuan Wu is a professor at Nanjing collage; Xiong You is an affiliate professor at Nanjing Agricultural collage; Bin Wang is a joint Ph.D pupil of Nanjing college and college of Cambridge.
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Extra resources for Structure-Preserving Algorithms for Oscillatory Differential Equations
4. 44) p+1 t∈ SN Tq . 3 Dispersion and Dissipation of RK(N) Methods It is known that when the RK(N) methods are used to solve oscillatory differential equations, they usually produce some dispersion and/or dissipation, even though these methods may be of high algebraic orders. 1 RK Methods Let us consider the simplest scalar first-order linear oscillatory equation y = iωy, ω > 0. 45) The true solutions of this equation satisfy y(xn + h) = eiν y(xn ), ν = hω. 46) These solutions are oscillatory with the frequency ω.
32) then become b¯s Mi = bs Ni , bs Ns = b¯s Ms . , φ1 (ν) − cs φ0 (ν) φ0 (ν) + ν 2 ci φ1 (ν) = φ0 (ν) + ν 2 cs φ1 (ν) φ1 (ν) − ci φ0 (ν) , which implies that (ci − cs ) φ02 (ν) + ν 2 φ12 (ν) = 0. From φ02 (ν) + ν 2 φ12 (ν) = 1, it follows that ci = cs , i = 1, . . , s − 1. 18) becomes bi φ0 (ν) + ν 2 Kφ1 (ν) a¯ ij = bj φ0 (ν) + ν 2 Kφ1 (ν) a¯ j i , for i, j = 1, . . , s. Since φ0 (ν) + ν 2 Kφ1 (ν) = 0, we have a¯ j i = i ≤ j. 6 The order of an SARKN method cannot exceed two. 36) i=1 s i=1 40 2 s ARKN Methods b¯i = φ2 (ν) + O(h).
The proof is complete. 38) where K= φ1 (ν) − 12 φ0 (ν) φ0 (ν) + 12 ν 2 φ1 (ν) . 38) is denoted SARKN1s2. 18) and one of the third-order conditions s bi a¯ ij = φ3 (ν) + O(h), i=1 we obtain a one-stage implicit SARKN method of order two: 1/2 1/6 b¯ Kφ1 (ν) b φ1 (ν) where K= φ1 (ν) − 12 φ0 (ν) φ0 (ν) + 12 ν 2 φ1 (ν) . 3 Phase and Stability Properties of Method SARKN1s2 In what follows, we are concerned with the phase and stability properties of the method SARKN1s2. For classical RKN methods, the stability properties are analyzed usually by applying the methods to the second-order homogeneous linear test model q¨ = −λ2 q, λ > 0.