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    A Lyapunov exponent based stability theory for ordinary differential equation initial value problem solvers

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    Issue Date
    2016-12-31
    Author
    Steyer, Andrew Jacob
    Publisher
    University of Kansas
    Format
    76 pages
    Type
    Dissertation
    Degree Level
    Ph.D.
    Discipline
    Mathematics
    Rights
    Copyright held by the author.
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    Abstract
    In this dissertation we consider the stability of numerical methods approximating the solution of bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees that a one-step method produces a decaying numerical solution to an asymptotically contracting, time-dependent, linear problem. This result is used to justify using a one-dimensional asymptotically contracting real-valued nonautonomous linear test problem to characterize the stability of a one-step method. The linear stability result is applied to prove a stability result for the numerical solution of a class of stable nonlinear problems. We use invariant manifold theory to show that we can obtain similar stability results for strictly stable linear multistep methods approximating asymptotically contracting, time-dependent, linear problems by relating their stability to the stability of an underlying one-step method. The stability theory for one-step methods is used to devise a procedure for stabilizing a solver that fails to produce a decaying solution to a linear problem when selecting step-size using standard error control techniques. Additionally, we develop an algorithm that selects step-size for the numerical solution of a decaying nonautonomous scalar test problem based on accuracy and the stability theory we developed.
    URI
    http://hdl.handle.net/1808/24193
    Collections
    • Mathematics Dissertations and Theses [180]
    • Dissertations [4472]

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    KU Libraries
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    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
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    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

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