Numerical artifacts in the generalized porous medium equation and solutions
Abstract/Contents
- Abstract
- The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to self-sharpening and its sharp corner solutions. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature for averaged-based second order finite volume discretizations. In these works, the numerical artifacts have been attributed to harmonic averaging of the coefficient k(p) for small p, and arithmetic averaging has been suggested as an alternative in the continuous coefficient case. In addition to the degeneracy and self-sharpening of the GPME with continuous coefficients, increased numerical challenges occur in the discontinuous coefficient case. These numerical challenges manifest themselves in spurious temporal oscillations in second order finite volume discretizations with both arithmetic and harmonic averaging. The integral average, developed in van der Meer et al. 2016, leads to improved solutions with monotone and reduced amplitude temporal oscillations. We aim to understand the cause of these artifacts, and to use that understanding to propose solutions. In the first part of this dissertation, we investigate the causes of the numerical artifacts using modified equation analysis for the continuous coefficient case. The provided modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. We find that the observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically in one and two dimensions through our newly developed Modified Harmonic Method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts. For the continuous coefficient case, we show results for two subclasses of the GPME with differentiable k(p) with respect to p, namely the Porous Medium Equation (PME) and the superslow diffusion equation. In the second part of this dissertation, we propose a new method called the Shock-Based Averaging Method (SAM) that incorporates the shock position into the numerical scheme. The shock position is numerically calculated by discretizing the theoretical speed of the front from the GPME theory. The speed satisfies the jump condition for integral conservation laws. SAM results in a non-oscillatory temporal profile, producing physically valid numerical results. We use SAM to demonstrate that the choice of averaging alone is not the cause of the oscillations, and that the shock position must be a part of the numerical scheme to avoid the artifacts. For the discontinuous coefficient case, we show results for a subclass of the discontinuous GPME, known as the Stefan problem.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2018; ©2018 |
| Publication date | 2018; 2018 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Maddix, Danielle Catherine |
|---|---|
| Degree supervisor | Gerritsen, Margot (Margot G.) |
| Thesis advisor | Gerritsen, Margot (Margot G.) |
| Thesis advisor | Sampaio, Luiz Augusto |
| Thesis advisor | Suckale, Jenny |
| Degree committee member | Sampaio, Luiz Augusto |
| Degree committee member | Suckale, Jenny |
| Associated with | Stanford University, Institute for Computational and Mathematical Engineering. |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Danielle Catherine Maddix. |
|---|---|
| Note | Submitted to the Institute for Computational and Mathematical Engineering. |
| Thesis | Thesis Ph.D. Stanford University 2018. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2018 by Danielle Catherine Maddix
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
Also listed in
Loading usage metrics...