WENO Computations and Pattern Stability

Doctoral Dissertation


There are two separated parts of my thesis. The first part is about how to design robust unstructured weighted essentially non-oscillatory (WENO) schemes. The WENO schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). A major difficulty for unstructured WENO is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. We combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme. The second part is about computational biology. We studied the pattern solutions of several mathematical models, chematical cell movement, Zebrafish Dorsalventral patterning and tumor angiogenesis. The detailed dynamics, pattern structures and robustness of the nonlinear reaction-diffusion model of chemotactic cell movements [89] are discussed using high resolution numerical simulations. We show that the model can form network patterns similar to early blood vessel structures seen experimentally. The model solutions do not blow up in finite time, a property of other chemotaxis cell movement models.

The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [128] has multiple steady state solutions. Seven steady state solutions are found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numerical simulations via a time marching approach. The results show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size. Similar approaches are applied to a more complicate free boundary problem on tumor growth.


Attribute NameValues
  • etd-05092012-144226

Author Yuan Liu
Advisor Yongtao Zhang
Contributor Yongtao Zhang, Committee Chair
Contributor Mark Alber, Committee Co-Chair
Contributor Andrew Sommese, Committee Member
Contributor Bei Hu, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Applied and Computational Mathematics and Statistics
Degree Name PhD
Defense Date
  • 2012-03-28

Submission Date 2012-05-09
  • United States of America

  • Conservation Law

  • Nonlinear Diffusion

  • Continuous Limit

  • Triangles

  • Unstructured Mesh

  • Zebrafish

  • Cellular Potts Model

  • Pattern Solution

  • WENO Scheme

  • Stability

  • Finite Difference/Volume Methods

  • Tumor Growth

  • Complex Geometry

  • Blood Vessel Formation

  • University of Notre Dame

  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units


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