Structure-Preserving Algorithms for Oscillatory Differential Equations
Xiong You
Language: English
Pages: 236
ISBN: 3642353371
Format: PDF / Kindle (mobi) / ePub
Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations by using theoretical analysis and numerical validation. Structure-preserving algorithms for differential equations, especially for oscillatory differential equations, play an important role in the accurate simulation of oscillatory problems in applied sciences and engineering. The book discusses novel advances in the ARKN, ERKN, two-step ERKN, Falkner-type and energy-preserving methods, etc. for oscillatory differential equations.
The work is intended for scientists, engineers, teachers and students who are interested in structure-preserving algorithms for differential equations. Xinyuan Wu is a professor at Nanjing University; Xiong You is an associate professor at Nanjing Agricultural University; Bin Wang is a joint Ph.D student of Nanjing University and University of Cambridge.
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i=1 s + h3 b¯i bj dfiJ ∧ dfjJ . (2.20) i,j =1 Differentiating the first equation of (2.17) yields s dQJi = dqnJ + ci h dq˙nJ + h2 a¯ ij dfjJ − ω2 dQJj , i = 1, . . . , s. a¯ ij dfjJ − ω2 dQJj , i = 1, . . . , s. j =1 Then, we have s dqnJ = dQJi − ci h dq˙nJ − h2 j =1 Therefore s a¯ ij dfjJ − ω2 dQJj ∧ dfiJ . dqnJ ∧ dfiJ = dQJi ∧ dfiJ − ci h dq˙nJ ∧ dfiJ − h2 j =1 (2.21) 2.2 Symplectic ARKN Methods 35 With (2.21), the formula (2.20) becomes J J dqn+1 ∧ dq˙n+1 s = dqnJ ∧
explicit ARKN methods of order five with the following Butcher tableau: c A¯ b¯ T (V ) bT (V ) = c1 c2 c3 c4 0 a¯ 21 a¯ 31 a¯ 41 0 0 a¯ 32 a¯ 42 0 0 0 a¯ 43 0 0 0 0 b¯1 (V ) b¯2 (V ) b¯3 (V ) b¯4 (V ) b1 (V ) b2 (V ) b3 (V ) b4 (V ) 2.3 Multidimensional ARKN Methods 59 The order conditions up to order five are eT ⊗ I b(V ) = φ1 (V ) + O h5 , cT ⊗ I b(V ) = φ2 (V ) + O h4 , c2 T ⊗ I b(V ) = 2φ3 (V ) + O h3 , ¯ T ⊗ I b(V ) = φ3 (V ) + O h3 , (Ae) c3 T ⊗ I b(V ) = 6φ4 (V ) + O h2 ,
) = q0 , q(t ˙ 0 ) = p0 , (4.1) where q : R → Rd and M ∈ Rd×d is a symmetric and positive semi-definite matrix implicitly containing the frequencies of the problem. When f (q) = −∇U (q) for some continuously differentiable function U (q), (4.1) is equivalent to a separable Hamiltonian system with the Hamiltonian H (p, q) = 12 p T p + 12 q T Mq + U (q). Since symplectic and symmetric methods have excellent structure-preserving behavior in long-term integration, it is natural to require the
Equations, DOI 10.1007/978-3-642-35338-3_6, © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2013 151 152 6 Adapted Falkner-Type Methods Later, implicit formulae were derived (see [2]): k yn+1 = yn + hyn + h 2 k βj∗ ∇ j gn+1 , γj∗ ∇ j gn+1 , yn+1 = yn + h j =0 (6.3) j =0 where the coefficients can be obtained from the generating functions ∞ Gβ ∗ (t) = βj∗ t j = t + (1 − t) ln(1 − t) ln (1 − t) 2 j =0 ∞ , Gγ ∗ (t) = γj∗ t j = j =0 −t . ln(1 − t) For the
used. The following discrete Gronwall’s lemma (Lemma 2.4 in [6]) is useful in the error analysis. Lemma 6.1 Suppose that α, ϕ, ψ and χ are nonnegative functions defined at xn = n x, n = 0, 1, . . . , N , and χ is nondecreasing. If k−1 ϕk + ψk ≤ χk + x αn ϕn , n=1 k = 0, 1, . . . , N, 6.3 Error Analysis 157 and if there is a constant cˆ such that k−1 αn ≤ c, ˆ x k = 0, 1, . . . , N, n=1 then ˆ ϕk + ψk ≤ χk eck x , k = 0, 1, . . . , N, where the subscript indices k and n denote the
