The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. In this work we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method to further reduce the computational cost; in total, three algorithms are developed. Initially a first order accurate in time scheme for low Reynolds number flows is considered. Next a second order algorithm useful for applications that require long-term time integration, such as climate and ocean forecasting is developed. Lastly, in order to extend this approach to convection dominated flows a model incorporating a POD spatial filter is presented. For all these schemes stability and convergence results for the ensemble-based methods are extended to the ensemble-POD schemes. Numerical results are provided to illustrate the theoretical stability and convergence results which have been proven. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / June 27, 2018. / ensemble computation, Navier-Stokes equations, Proper Orthogonal Decomposition / Includes bibliographical references. / Max Gunzburger, Professor Directing Thesis; Mark Sussman, University Representative; Janet Peterson, Committee Member; Gordon Erlebacher, Committee Member; Chen Huang, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_647298 |
| Contributors | Schneier, Michael (author), Gunzburger, Max D. (professor directing thesis), Sussman, Mark (university representative), Peterson, Janet S. (committee member), Erlebacher, Gordon, 1957- (committee member), Huang, Chen (committee member), Florida State University (degree granting institution), College of Arts and Sciences (degree granting college), Department of Scientific Computing (degree granting departmentdgg) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text, doctoral thesis |
| Format | 1 online resource (80 pages), computer, application/pdf |
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