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Date of Award

2019

Document Type

Campus Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Zhengfu Xu

Committee Member 1

Jingfeng Jiang

Committee Member 2

Alexander E. Labovsky

Committee Member 3

Jiguang Sun

DOI

10.37099/mtu.dc.etdr/842

Abstract

This dissertation includes four Chapters. A brief description about each chapter is organized as follows.

In Chapter 1, I will give some necessary definitions, important theoretical and numerical results related to the next few chapters about scalar hyperbolic conservation laws. Also some numerical difficulties as well as the motivation for constructing total variation bounded (TVB) flux limiter discussed more in Chapters 2 and Chapter 3 will be given.

In Chapter 2, for solving 1D scalar conservation laws, a provable total variation bounded flux limiter high order finite difference methods based on variation measured on grid values will be discussed. A new criterion will be provided to design TVB flux limiter for high order (at least third order) finite difference schemes for one-dimensional scalar conservation laws. When this TVB flux limiter is applied to a third order finite difference scheme, we show that the third order of accuracy is maintained from the local truncation error point of view. Numerical results are also produced to demonstrate: the total variation of the numerical solution is always bounded by the initial total variation; the order of accuracy is not sacrificed. This is a published paper.

In Chapter 3, some important and useful achievements about constructing a proper definition of numerical total variation in high dimensional spaces will be presented. It would be used for a further analysis and future publication for total variation bounded schemes for 2- and higher dimensional scalar conservation laws. This is an unpublished research material that tries to generate the idea/approach proposed in Chapter 2 to higher dimensional spaces.

In Chapter 4, to improve the denosing performance on 3D biomedical images, a mathematical model based on both spaces (divergence and curl) and time (consistency) is analyzed, and the Polak-Ribiere-Polyak conjugate gradient method associated with the technique of golden-section search for 1D minimization under the framework of the multiplier method of Rochafellar is applied. Although the main algorithm therein is not original, some modifications and idea made upon part of the algorithm are original by the authors.

A list of References and Appendixes related to my research work will be given in the end to enclose this dissertation.

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