Colorado School of Mines
Orthogonal spline collocation methods for fluid flow problems.
Degree: PhD, Applied Mathematics and Statistics, 2019, Colorado School of Mines
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 H1
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 H1
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.
Advisors/Committee Members: Bialecki, Bernard (advisor), Martin, Paul (committee member), Fasshauer, Greg (committee member), Tilton, Nils (committee member).
Subjects/Keywords: Burgers' Equation; Navier-Stokes Equations; Poisson's Equation; Matrix Decomposition Algorithm; Alternating Direction Implicit; Orthogonal Spline Collocation
to Zotero / EndNote / Reference
APA (6th Edition):
Fisher, N. (2019). Orthogonal spline collocation methods for fluid flow problems. (Doctoral Dissertation). Colorado School of Mines. Retrieved from http://hdl.handle.net/11124/173048
Chicago Manual of Style (16th Edition):
Fisher, Nicholas. “Orthogonal spline collocation methods for fluid flow problems.” 2019. Doctoral Dissertation, Colorado School of Mines. Accessed June 25, 2019.
MLA Handbook (7th Edition):
Fisher, Nicholas. “Orthogonal spline collocation methods for fluid flow problems.” 2019. Web. 25 Jun 2019.
Fisher N. Orthogonal spline collocation methods for fluid flow problems. [Internet] [Doctoral dissertation]. Colorado School of Mines; 2019. [cited 2019 Jun 25].
Available from: http://hdl.handle.net/11124/173048.
Council of Science Editors:
Fisher N. Orthogonal spline collocation methods for fluid flow problems. [Doctoral Dissertation]. Colorado School of Mines; 2019. Available from: http://hdl.handle.net/11124/173048