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scaling and squaring technique where the Padé or Taylor methods to compute the exponential of the scaled matrix have been replaced by appropriate splitting methods tailored for this class of matrices.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 ) 1 1.5 2 2.5 3 3.5 4 εε ε εεεε 4.5 0 0.2 0.4 0.6 0.8 1 aa 2 aa 1 Squaring (4,2) Padé rr 2 Padé rr 4 Padé rr 2 squared 1 1.5 2 2.5 3 3.5 4 εε ε εεεεε 4.5 0 0.2 0.4 0.6 0.8 1 aa 2 Author: Muaz Seydaoğlu Advisors: Prof. Dr. Sergio Blanes Zamora Dr. Philipp Bader Valencia, September 2016
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
Summary
Resumen
Resum
1 Motivation 1 2 Introduction 5 3 Non-reversible systems 33 4 Exponential of perturbed matrices 49
5 Matrix Hill s equation 69 6 Conclusions 89 A Algebraic Tools 93 References 97
2.1 Splitting methods
Vertauschungssatz
Symplectic Euler and The Störmer-Verlet method:
2.2 Composition 2.2.1 Construction of higher order integrators by composition
2.2.2 Lie derivative and integrators
2.3 Order conditions via BCH formula
2.3.1 Runge-Kutta-Nyström methods (RKN)
2.3.2 Near-integrable systems
2.4 The Magnus Expansion (ME)
2.4.1 Derivative of the exponential and its inverse
Lemma 2.4.1. [18] The derivative of a matrix exponential given by (1) (2) (3) Lemma 2.4.2. [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 2.4.3. [18] The solution of the differential equation with initial condition can be written as with defined by
2.4.2 First few terms of the Magnus expansion
2.4.3 Time symmetry of the Magnus expansion 2.4.4 Convergence of the Magnus expansion
Theorem 2.4.4. [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 ). 2.4.5 Numerical integrators via Magnus expansion
Fourth Order Magnus Integrator: Sixth Order Magnus Integrator: Eight Order Magnus Integrator:
2.4.6 Commutator-Free Magnus integrators
2.4.7 Different time-averaging
Near-seperable systems : do enddo
3.1 The separable non-autonomous parabolic equations
3.1.1 The problem parabolic heat equation
Lie-Trotter splitting symmetrized Strang splitting triple-jump composition
3.2 Splitting methods for non-autonomous problems
3.2.1 Splitting methods for non-autonomous perturbed systems
3.2.2 Order conditions
3.2.3 Fourth-order methods
3.3 Numerical examples
Example 1
Example 2: A linear parabolic equation.
Example 3: The semi-linear reaction-diffusion equation of Fisher.
scaling and squaring method 4.1 The scaling, splitting and squaring method
adiabatic picture
4.2 Computational cost of matrix exponentiation 4.2.1 Computational cost of Taylor and Padé methods Taylor methods
Diagonal Padé methods 4.2.2 Computational cost of splitting methods
1 1 1 2 4.3 The Lie algebra of a perturbed system: methods
4.3.1 Error propagation by squaring processor
kernel Theorem 4.3.1. 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.
Example 4.4 Splitting methods for scaling and squaring
4.4.1 Standard splittings
Modified squarings
4.4.2 Modified splittings
4.5 Error analysis
5.37 2.51 2.48 3.85 C C
4.6 Numerical results 4.6.1 Rotations
4.6.2 Dissipation
5.1 Symplectic integrators for the matrix Hill s equation
5.2 Numerical integration for one period
5.2.1 Symplectic methods Implicit symplectic Runge Kutta methods
Splitting methods do enddo
Magnus integrators
The computational cost of exponential of matrices
Commutator-free Magnus integrators 5.3 Exponential symplectic methods for the Hill s equation 5.3.1 Sixth-order methods
One- -exponential method
Two- -exponential method first commutes with last Three- -exponential method
5.3.2 Eigth-order methods Five- -exponential method
5.4 Numerical examples
5.4.1 The Mathieu equation
log 10 λi 1 log 10 λi 1 log 10 λi 1 log 10 λi 1 2 0 2 RK [6] 7 4 0 1 2 3 4 5 ω 5 0 5 10 RKN [6] 11 15 0 1 2 3 4 5 ω 0 5 10 14 0 5 10 14 log 10 λi 1 log 10 λi 1 5 0 5 10 RKGL [6] 15 0 1 2 3 4 5 ω 5 0 5 10 Φ [6] 2 15 0 1 2 3 4 5 ω 0 RK [6] 7 5.013 5.0134 5.0138 ω RKN [6] 11 5.013 5.0134 5.0138 ω log 10 λi 1 log 10 λi 1 5 10 14 0 5 10 14 RKGL [6] 5.013 5.0134 5.0138 ω Φ [6] 2 5.013 5.0134 5.0138 ω 5.4.2 Matrix Hill s equation
log 10 ( error 1) 0 2 4 6 RK [6] 7 RKGL [6] RKN [6] 11 Φ [6] 2 8 0 2 4 6 8 10 ω log 10 ( error 1) 1 2 3 4 5 6 7 8 RK [6] 7 RKGL [6] RKN [6] 11 Φ [6] 5 Φ [6] 1 Φ [6] 2 Φ [6] 3 Φ [8] 5 2 2.2 2.4 2.6 2.8 log 10 (cost)
r = 5, ε = 5 r = 5, ε = 1/2 0 0 2 2 log 10 ( error 1) 4 6 8 10 RK [6] 7 RKGL [6] RKN [6] 11 2.5 3 3.5 log 10 ( error 1) 4 6 8 10 2.5 3 3.5 log 10 (cost) log 10 (cost) r = 7, ε = 7 r = 7, ε = 7/10 0 0 2 log 10 ( error 1) 5 10 Φ [6] 5 Φ [6] 1 Φ [6] 2 Φ [6] 3 Φ [8] 5 2.5 3 3.5 log 10 ( error 1) 4 6 8 10 3 3.5 4 4.5 log 10 (cost) log 10 (cost) 5.4.3 The damped Mathieu equation
5.4.4 The non-linear Mathieu equation
6.1 Non-reversible systems
6.2 Exponential of perturbed matrices 6.3 Symplectic integrators for the matrix Hill s equation
A.1 Further approaches A.1.1 On processing processing techniques
A.1.2 More exponentials A.1.3 Splitting for low-order Padé
A.2 Efficient symplectic approximation of
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