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Adaptive Exponential Runge–Kutta Pairs for Stiff Differential Equations

  • Author / Creator
    Zoto, Thoma
  • Numerical methods are one of the most important aspects in the computer modelling of physical phenomena governed by differential equations. In order to study these phenomena precisely and in a timely manner, we need to use robust and efficient numerical methods for solving the underlying differential equations. The aim of exponential integrator methods is to provide robustness when solving stiff differential equations and the aim of this work is not only to construct new exponential integrators, but also to introduce ideas that allow for efficient implementation of such methods.
    An extensive discussion on stiffness, concluding with a definition of numerical stiffness, is given in Chapter 2. Exponential integrator methods as well as the order conditions that need to be satisfied by them are presented in Chapter 3. Embedded methods and step-size adjustment is discussed in Chapter 4, together with a new optimized embedded exponential integrator method denoted ERK43ZB. In Chapter 5 we present a novel idea involving the Schur decomposition that allows for a substantial increase in the efficiency and memory usage of exponential integrators, in particular embedded exponential integrators. The thesis ends with two Mathematica scripts given in Appendix A and B that are used to check the order of an exponential method and to systematically construct new optimized embedded exponential methods, respectively.

  • Subjects / Keywords
  • Graduation date
    Fall 2022
  • Type of Item
    Thesis
  • Degree
    Master of Science
  • DOI
    https://doi.org/10.7939/r3-qa03-vg34
  • License
    This thesis is made available by the University of Alberta Library 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.