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A Lyapunov exponent based stability theory for ordinary differential equation initial value problem solvers
dc.contributor.advisor | Van Vleck, Erik S | |
dc.contributor.author | Steyer, Andrew Jacob | |
dc.date.accessioned | 2017-05-15T22:15:33Z | |
dc.date.available | 2017-05-15T22:15:33Z | |
dc.date.issued | 2016-12-31 | |
dc.date.submitted | 2016 | |
dc.identifier.other | http://dissertations.umi.com/ku:14912 | |
dc.identifier.uri | http://hdl.handle.net/1808/24193 | |
dc.description.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. | |
dc.format.extent | 76 pages | |
dc.language.iso | en | |
dc.publisher | University of Kansas | |
dc.rights | Copyright held by the author. | |
dc.subject | Mathematics | |
dc.subject | differential equations | |
dc.subject | Lyapunov exponent | |
dc.subject | numerical analysis | |
dc.subject | ODE | |
dc.subject | Runge-Kutta | |
dc.title | A Lyapunov exponent based stability theory for ordinary differential equation initial value problem solvers | |
dc.type | Dissertation | |
dc.contributor.cmtemember | Huang, Weizhang | |
dc.contributor.cmtemember | Liu, Weishi | |
dc.contributor.cmtemember | Xu, Hongguo | |
dc.contributor.cmtemember | Mechem, David | |
dc.thesis.degreeDiscipline | Mathematics | |
dc.thesis.degreeLevel | Ph.D. | |
dc.identifier.orcid | ||
dc.rights.accessrights | openAccess |
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Mathematics Dissertations and Theses [179]