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1. Liu, Kun. Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data.

Degree: PhD, Engineering, 2013, Rice University

URL: http://hdl.handle.net/1911/71989

This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates M realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed.
*Advisors/Committee Members: Riviere, Beatrice M. (advisor), Heinkenschloss, Matthias (committee member), Symes, William W. (committee member), Vannucci, Marina (committee member).*

Subjects/Keywords: Parabolic PDEs; Monte Carlo Discontinuous Galerkin; Locally mass conservation; Random input data; Kernel PCA; Random permeability; Darcy's Law; Coupled flow and transport

…combining a random sampling
technique and a *locally* *mass* conservative method, the uncertainty of… …oscillations in numerical simulations. However, these methods do not possess local
*mass* *conservation*… …However, it does not
satisfy local *mass* *conservation*, which is a crucial property in reservoir… …violation of
the law of local *mass* *conservation* in velocity fields could result in “spurious… …attractive advantages of DG
methods include: local *mass* *conservation*, complex geometrics, high…

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APA (6^{th} Edition):

Liu, K. (2013). Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data. (Doctoral Dissertation). Rice University. Retrieved from http://hdl.handle.net/1911/71989

Chicago Manual of Style (16^{th} Edition):

Liu, Kun. “Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data.” 2013. Doctoral Dissertation, Rice University. Accessed November 12, 2019. http://hdl.handle.net/1911/71989.

MLA Handbook (7^{th} Edition):

Liu, Kun. “Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data.” 2013. Web. 12 Nov 2019.

Vancouver:

Liu K. Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data. [Internet] [Doctoral dissertation]. Rice University; 2013. [cited 2019 Nov 12]. Available from: http://hdl.handle.net/1911/71989.

Council of Science Editors:

Liu K. Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data. [Doctoral Dissertation]. Rice University; 2013. Available from: http://hdl.handle.net/1911/71989