SANGRADO PAGINA 17CMX24CM. PhD Thesis. Splitting methods for autonomous and non-autonomous perturbed equations LOMO A AJUSTAR (AHORA 4CM)

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1 SANGRADO A3 PAGINA 17CMX24CM LOMO A AJUSTAR (AHORA 4CM) We have considered the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients. A straightforward application of splitting methods with complex coefficients to non-autonomous problems require the evaluation of the time-dependent functions in the operators at complex times, and the corresponding flows in the numerical scheme are, in general, not well conditioned. To circumvent this difficulty, in this work we study a class of methods in which one set of the coefficients belongs to the class of real and positive numbers. Taking the time as a new coordinate, an appropriate splitting of the system allows us to build numerical schemes, where all timedependent operators are evaluated at real values of the time. This technique is promising for perturbed systems which are analyzed in more detail. We have proposed a new recursive algorithm based on splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+εB of a sparse and efficiently exponentiable matrix D with sparse exponential exp(d) and a dense matrix εb which is of small norm in comparison with D. The algorithm is based on the scaling and squaring technique where the Padé or Taylor methods to compute the exponential of the scaled matrix have been replaced by appropriate splitting methods tailored for this class of matrices. We have studied the numerical integration of the matrix Hill equation using methods that accurately reproduce the parametric resonances of the exact solution. We are mainly interested in the Hamiltonian case which is the most frequent one in practice, namely when Hill s equations originate from a Hamiltonian function. In this case the fundamental matrix solution is a symplectic matrix and we illustrate the importance of the preservation of this property by the numerical integrators. Referees: Prof. Dr. Begoña Cano Urdiales, Universidad de Valladolid Prof. Dr. Cesáreo González Fernández, Universidad de Valladolid Prof. Dr. Severiano González Pinto, Universidad de La Laguna Jury: Prof. Dr. Juan Ignacio Montijano Torcal, Universidad de Zaragoza Prof. Dr. Damián Ginestar Peiro, Universidad Politécnica de Valencia Prof. Dr. Severiano González Pinto, Universidad de La Laguna Muaz SEYDAOĞLU SPLITTING METHODS FOR AUTONOMOUS AND NON-AUTONOMOUS PERTURBED EQUATIONS PhD Thesis / 2016 PhD Thesis Splitting methods for autonomous and non-autonomous perturbed equations log 10 ( exp(aaa A AAAAAAA ex ) 2 ) εε ε εεεε aa 2 aa 1 Squaring (4,2) Padé rr 2 Padé rr 4 Padé rr 2 squared εε ε εεεεε aa 2 Author: Muaz Seydaoğlu Advisors: Prof. Dr. Sergio Blanes Zamora Dr. Philipp Bader Valencia, September 2016

2 Splitting methods for autonomous and non-autonomous perturbed equations PhD Thesis Author: Muaz Seydaoğlu Advisors: Prof. Dr. Sergio Blanes Zamora Dr. Philipp Bader Valencia, September 2016

6 Summary

8 Resumen

10 Resum

12 1 Motivation 1 2 Introduction 5 3 Non-reversible systems 33 4 Exponential of perturbed matrices 49

13 5 Matrix Hill s equation 69 6 Conclusions 89 A Algebraic Tools 93 References 97

18 2.1 Splitting methods

19 Vertauschungssatz

20 Symplectic Euler and The Störmer-Verlet method:

21 2.2 Composition Construction of higher order integrators by composition

23 2.2.2 Lie derivative and integrators

26 2.3 Order conditions via BCH formula

28 2.3.1 Runge-Kutta-Nyström methods (RKN)

29 2.3.2 Near-integrable systems

30 2.4 The Magnus Expansion (ME)

31 2.4.1 Derivative of the exponential and its inverse

32 Lemma [18] The derivative of a matrix exponential given by (1) (2) (3) Lemma [9] is invertible if the eigenvalues of the linear operator are different from with. In addition, if then, a convergent expansion is given as where are the Bernoulli numbers, defined by Theorem [18] The solution of the differential equation with initial condition can be written as with defined by

33 2.4.2 First few terms of the Magnus expansion

34 2.4.3 Time symmetry of the Magnus expansion Convergence of the Magnus expansion

35 Theorem [31] Let be a bounded operator in a Hilbert space for the differential equation with. The Magnus series, with given by convergences on the interval such that and. The statement also remains valid with a normal operator (in particular, with unitary ) Numerical integrators via Magnus expansion

39 Fourth Order Magnus Integrator: Sixth Order Magnus Integrator: Eight Order Magnus Integrator:

40 2.4.6 Commutator-Free Magnus integrators

41 2.4.7 Different time-averaging

44 Near-seperable systems : do enddo

46 3.1 The separable non-autonomous parabolic equations

47 3.1.1 The problem parabolic heat equation

48 Lie-Trotter splitting symmetrized Strang splitting triple-jump composition

49 3.2 Splitting methods for non-autonomous problems

50 3.2.1 Splitting methods for non-autonomous perturbed systems

52 3.2.2 Order conditions

54 3.2.3 Fourth-order methods

55 3.3 Numerical examples

56 Example 1

57 Example 2: A linear parabolic equation.

58 Example 3: The semi-linear reaction-diffusion equation of Fisher.

62 scaling and squaring method 4.1 The scaling, splitting and squaring method

63 adiabatic picture

65 4.2 Computational cost of matrix exponentiation Computational cost of Taylor and Padé methods Taylor methods

66 Diagonal Padé methods Computational cost of splitting methods

67 The Lie algebra of a perturbed system: methods

68 4.3.1 Error propagation by squaring processor

69 kernel Theorem Let a diagonalizable matrix and let be an -stage splitting method of order that approximates the scaled exponential with. Then, for sufficiently small values of and we have that where are constants which depend on the norm but neither on nor on. Proof.

70 Example 4.4 Splitting methods for scaling and squaring

71 4.4.1 Standard splittings

72 Modified squarings

74 4.4.2 Modified splittings

75 4.5 Error analysis

76 C C

78 4.6 Numerical results Rotations

79 4.6.2 Dissipation

82 5.1 Symplectic integrators for the matrix Hill s equation

85 5.2 Numerical integration for one period

86 5.2.1 Symplectic methods Implicit symplectic Runge Kutta methods

87 Splitting methods do enddo

88 Magnus integrators

89 The computational cost of exponential of matrices

90 Commutator-free Magnus integrators 5.3 Exponential symplectic methods for the Hill s equation Sixth-order methods

91 One- -exponential method

92 Two- -exponential method first commutes with last Three- -exponential method

93 5.3.2 Eigth-order methods Five- -exponential method

94 5.4 Numerical examples

95 5.4.1 The Mathieu equation

96 log 10 λi 1 log 10 λi 1 log 10 λi 1 log 10 λi RK [6] ω RKN [6] ω log 10 λi 1 log 10 λi RKGL [6] ω Φ [6] ω 0 RK [6] ω RKN [6] ω log 10 λi 1 log 10 λi RKGL [6] ω Φ [6] ω Matrix Hill s equation

97 log 10 ( error 1) RK [6] 7 RKGL [6] RKN [6] 11 Φ [6] ω log 10 ( error 1) RK [6] 7 RKGL [6] RKN [6] 11 Φ [6] 5 Φ [6] 1 Φ [6] 2 Φ [6] 3 Φ [8] log 10 (cost)

98 r = 5, ε = 5 r = 5, ε = 1/ log 10 ( error 1) RK [6] 7 RKGL [6] RKN [6] log 10 ( error 1) log 10 (cost) log 10 (cost) r = 7, ε = 7 r = 7, ε = 7/ log 10 ( error 1) 5 10 Φ [6] 5 Φ [6] 1 Φ [6] 2 Φ [6] 3 Φ [8] log 10 ( error 1) log 10 (cost) log 10 (cost) The damped Mathieu equation

100 5.4.4 The non-linear Mathieu equation

101

102 6.1 Non-reversible systems

103 6.2 Exponential of perturbed matrices 6.3 Symplectic integrators for the matrix Hill s equation

104

105

106 A.1 Further approaches A.1.1 On processing processing techniques

107 A.1.2 More exponentials A.1.3 Splitting for low-order Padé

108 A.2 Efficient symplectic approximation of

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THE Hamilton equations of motion constitute a system

THE Hamilton equations of motion constitute a system Proceedings of the World Congress on Engineering 0 Vol I WCE 0, July 4-6, 0, London, U.K. Systematic Improvement of Splitting Methods for the Hamilton Equations Asif Mushtaq, Anne Kværnø, and Kåre Olaussen