By Xinyuan Wu, Kai Liu, Wei Shi
This e-book describes various powerful and effective structure-preserving algorithms for second-order oscillatory differential equations. Such structures 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 effectively simulate the genuine habit of such structures, a numerical set of rules needs to shield up to attainable their key structural homes: time-reversibility, oscillation, symplecticity, and effort and momentum conservation. The e-book describes novel advances in RKN tools, ERKN tools, Filon-type asymptotic equipment, AVF equipment, and trigonometric Fourier collocation methods. The accuracy and potency of every of those algorithms are proven through cautious numerical simulations, and their structure-preserving houses are carefully verified by means of theoretical research. The ebook additionally offers insights into the sensible implementation of the methods.
This publication is meant for engineers and scientists investigating oscillatory structures, in addition to for lecturers 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
Example text
A ¯ (V ) . . a ¯ (V ) a (V ) . . a (V ) c s s1 ss s1 ss b¯ (V ) b (V ) b¯1 (V ) . . b¯s (V ) b1 (V ) . . 46) has the form ⎧ s ⎪ ⎪ 2 2 2 ⎪ = φ (c V )y + c φ (c V )hy + h a¯ i j (V ) f (xn + c j h, Y j , Y j ), Y i 0 n i 1 ⎪ n i i ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ s ⎪ ⎪ ⎪ 2 2 ⎪ Y = −c h Mφ (c V )y + φ (c V )y + h ai j (V ) f (xn + c j h, Y j , Y j ), ⎪ i 1 n 0 n i i ⎪ i ⎨ i = 1, . . , s, i = 1, . . , s, j=1 s ⎪ ⎪ ⎪ ⎪ yn+1 = φ0 (V )yn + φ1 (V )hyn + h 2 b¯i (V ) f (xn + ci h, Yi , Yi ), ⎪ ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ s ⎪ ⎪ ⎪ ⎪ bi (V ) f (xn + ci h, Yi , Yi ).
8 Towards ERKN Methods for General Second-Order Oscillatory Systems 17 is the stepsize, and yn and yn are approximations to the values of y(x) and y (x) at xn = x0 + nh, respectively, for n = 1, 2, . .. This method can also be represented compactly in Butcher’s tableau of coefficients: c1 a¯ 11 (V ) · · · a¯ 1s (V ) a11 (V ) · · · a1s (V ) .. .. .. .. ¯ ) A(V ) c A(V . . . . = a ¯ (V ) . . a ¯ (V ) a (V ) . . a (V ) c s s1 ss s1 ss b¯ (V ) b (V ) b¯1 (V ) . . b¯s (V ) b1 (V ) . . 46) has the form ⎧ s ⎪ ⎪ 2 2 2 ⎪ = φ (c V )y + c φ (c V )hy + h a¯ i j (V ) f (xn + c j h, Y j , Y j ), Y i 0 n i 1 ⎪ n i i ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ s ⎪ ⎪ ⎪ 2 2 ⎪ Y = −c h Mφ (c V )y + φ (c V )y + h ai j (V ) f (xn + c j h, Y j , Y j ), ⎪ i 1 n 0 n i i ⎪ i ⎨ i = 1, .
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 [63]) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s Yi = yn + ci hyn + h 2 a¯ i j f (tn + c j h, Y j ) − MY j , i = 1, 2, .