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dc.contributor.advisorBialecki, Bernard
dc.contributor.authorFisher, Nicholas Llewellyn
dc.date.accessioned2019-06-04T14:20:36Z
dc.date.accessioned2022-02-03T13:16:34Z
dc.date.available2019-06-04T14:20:36Z
dc.date.available2022-02-03T13:16:34Z
dc.date.issued2019
dc.identifierFisher_mines_0052E_11719.pdf
dc.identifierT 8710
dc.identifier.urihttps://hdl.handle.net/11124/173048
dc.descriptionIncludes bibliographical references.
dc.description2019 Spring.
dc.description.abstractWe propose an approach for the numerical solution of the Navier-Stokes equations based on a pressure Poisson equation reformulation. Through an alternating direction implicit extrapolated Crank--Nicolson time discretization, the scheme decouples the updates for velocity and pressure terms. Moreover, the proposed scheme reduces the Navier-Stokes equations to a Burgers' equation for the velocity terms and a singular Neumann Poisson equation for the pressure. These two sub-problems are analyzed in turn. We use extrapolated alternating direction implicit Crank-Nicolson orthogonal spline collocation with splines of order $r$ to solve the coupled Burgers' equations in two space variabl and two unknown functions. The scheme is initialized with an alternating direction implicit predictor-corrector method. We show theoretically that the $H^1$ norm of the error at each time level is of order $r$ in space and of order 2 in time. Then we use a matrix decomposition algorithm for the orthogonal spline collocation solution to Poisson's equation with Neumann boundary conditions. We show theoretically that the $H^1$ semi-norm of the error is of order $r$. In each case, our numerical results confirm these theoretical orders. Finally, the combined scheme is implemented for the solution of the pressure Poisson reformulation of the Navier--Stokes equations using splines of equal order. Numerical results show that the scheme obtains the expected optimal order convergence rates for both the velocity and pressure terms.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado School of Mines. Arthur Lakes Library
dc.rightsCopyright of the original work is retained by the author.
dc.subjectBurgers' equation
dc.subjectNavier-Stokes equations
dc.subjectPoisson's equation
dc.subjectmatrix decomposition algorithm
dc.subjectalternating direction implicit
dc.subjectorthogonal spline collocation
dc.titleOrthogonal spline collocation methods for fluid flow problems
dc.typeText
dc.contributor.committeememberMartin, P. A.
dc.contributor.committeememberFasshauer, Gregory
dc.contributor.committeememberTilton, Nils
thesis.degree.nameDoctor of Philosophy (Ph.D.)
thesis.degree.levelDoctoral
thesis.degree.disciplineApplied Mathematics and Statistics
thesis.degree.grantorColorado School of Mines


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