Orthogonal spline collocation methods for fluid flow problems
dc.contributor.advisor | Bialecki, Bernard | |
dc.contributor.author | Fisher, Nicholas Llewellyn | |
dc.date.accessioned | 2019-06-04T14:20:36Z | |
dc.date.accessioned | 2022-02-03T13:16:34Z | |
dc.date.available | 2019-06-04T14:20:36Z | |
dc.date.available | 2022-02-03T13:16:34Z | |
dc.date.issued | 2019 | |
dc.identifier | Fisher_mines_0052E_11719.pdf | |
dc.identifier | T 8710 | |
dc.identifier.uri | https://hdl.handle.net/11124/173048 | |
dc.description | Includes bibliographical references. | |
dc.description | 2019 Spring. | |
dc.description.abstract | We 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.medium | born digital | |
dc.format.medium | doctoral dissertations | |
dc.language | English | |
dc.language.iso | eng | |
dc.publisher | Colorado School of Mines. Arthur Lakes Library | |
dc.rights | Copyright of the original work is retained by the author. | |
dc.subject | Burgers' equation | |
dc.subject | Navier-Stokes equations | |
dc.subject | Poisson's equation | |
dc.subject | matrix decomposition algorithm | |
dc.subject | alternating direction implicit | |
dc.subject | orthogonal spline collocation | |
dc.title | Orthogonal spline collocation methods for fluid flow problems | |
dc.type | Text | |
dc.contributor.committeemember | Martin, P. A. | |
dc.contributor.committeemember | Fasshauer, Gregory | |
dc.contributor.committeemember | Tilton, Nils | |
thesis.degree.name | Doctor of Philosophy (Ph.D.) | |
thesis.degree.level | Doctoral | |
thesis.degree.discipline | Applied Mathematics and Statistics | |
thesis.degree.grantor | Colorado School of Mines |