Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.

Padgett, Josh Lee, 1990-

Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.

dc.contributor.advisor	Sheng, Qin (Tim)
dc.creator	Padgett, Josh Lee, 1990-
dc.date.accessioned	2017-09-28T12:55:58Z
dc.date.available	2017-09-28T12:55:58Z
dc.date.created	2017-08
dc.date.issued	2017-07-08
dc.date.submitted	August 2017
dc.date.updated	2017-09-28T12:55:59Z
dc.description.abstract	In this dissertation, we explore and analyze highly effective and efficient computational procedures for solving a class of nonlinear and stochastic partial differential equations. We are particularly interested in degenerate Kawarada equations arising from numerous multiphysics applications including solid fuel combustion and oil pipeline decay. These differential equations are characterized by their strong degeneracies on the boundary, quenching blow-up singularities in the spatial domain, and vibrant stochastic influences in the evolution processes. In addition, physically relevant numerical solutions to Kawarada-type equations must preserve their positivity and monotonicity in time throughout computations, for all valid initial datum. In light of these concerns and challenges, the numerical methods and analysis developed in this dissertation incorporate nonuniform finite difference approximations of the underlying singular differential equation on arbitrary spatial grids. These approximations are then advanced in time via proper adaptive mechanisms and splitting procedures, when multiple dimensions are involved. Degenerate stochastic Kawarada equations in one-, two-, and three-dimensions are explored. The numerical procedures developed in each case are rigorously shown to not only be unconditionally stable, but also preserve the anticipated solution positivity and monotonicity throughout computations. While the numerical stability is proven in the local linearized sense, the stability analysis has been successfully extended to a novel setting that significantly considers all nonlinear influences. Nonlinear convergence of the quenching numerical solutions, as well as their temporal derivatives, are discussed and assessed through generalized Milne devices for the two-dimensional problem. These explorations offer new insights into the computations of similar singular and stochastic differential equation problems. Intensive simulation experiments and validations are provided.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/2104/10099
dc.language.iso	en
dc.rights.accessrights	Worldwide access.
dc.rights.accessrights	Access changed 12/11/19.
dc.subject	Partial differential equations. Operator splitting methods.
dc.title	Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.
dc.type	Thesis
dc.type.material	text
local.embargo.lift	2019-08-01
local.embargo.terms	2019-08-01
thesis.degree.department	Baylor University. Dept. of Mathematics.
thesis.degree.grantor	Baylor University
thesis.degree.level	Doctoral
thesis.degree.name	Ph.D.