Uncertainty Quantification of Dynamical Systems and Stochastic Symplectic Schemes

• Author / Creator

  Deng, Jian

• It has been known that for some physical problems, a small change in the system parameters or in the initial/boundary conditions could leas to a significant change in the system response. Hence, it is of importance to investigate the impact of uncertainty on dynamical system in order to fully understand the system behavior. In this thesis, numerical methods used to simulate the effect of random/stochastic perturbation on dynamical systems are studied. In the first part of this thesis, an aeroelastic system model representing an oscillating airfoil in pitch and plunge with random variations in the flow speed, the structural stiffness terms and initial conditions are concerned. Two approaches, stochastic normal form and stochastic collocation method, are proposed to investigate the Hopf bifurcation and the secondary bifurcation behavior, respectively. Stochastic normal form allows us to study analytically the Hopf bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation; while numerical simulations demonstrate the effectiveness of stochastic collocation method for long term computation and discontinuous problems. In the second part of this work, we focus the construction of efficient and robust computational schemes for stochastic system, and the stochastic symplectic schemes for stochastic Hamiltonian system are developed. A systematic procedure to construct symplectic numerical schemes for stochastic Hamiltonian systems is presented. The approach is an extension to the stochastic case of the methods based on generating functions. The idea is also extended to the symplectic weak scheme construction. Theoretical analysis of the convergence is reported for strong/weak symplectic integrators. The numerical simulations are carried out to confirm that the symplectic methods are efficient computational tools
  for long-term behaviors. Moreover, the coefficients of the generating function are invariant under permutations for the stochastic Hamiltonian system preserving Hamiltonian functions. As a consequence the high-order symplectic weak and strong methods have simpler forms than the Taylor expansion schemes with the same order.

• Subjects / Keywords

  • Stochastic differential equations
  • Uncertainty Quantification
  • Stochastic symplectic integrator

• Graduation date

  Spring 2013

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R36P6W

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.