Date of Award

8-31-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

David Shirokoff

Second Advisor

Wooyoung Choi

Third Advisor

Richard O. Moore

Fourth Advisor

Yassine Boubendir

Fifth Advisor

Benjamin Seibold

Abstract

This dissertation focuses on developing efficient and stable (high order) time-stepping strategies for the dispersive shallow water equations (DSWE) with variable bathymetry. The DSWE extends the regular shallow water equations to include dispersive effects. Dispersion is physically important and can maintain the shape of a wave that would otherwise form a shock in the shallow water system.

In some cases, the DSWE may be simplified when the bathymetry length scales are small (or large) in relation to other length scales in the shallow water system. These simplified DSWE models, which are related to the full DSWEs, are also considered in this thesis as well.

Incorporating dispersive effects creates added difficulties when devising efficient high order time-stepping methods. Time-stepping the DSWE is difficult as the equations may be stiff as well as non-local and nonlinear in the time derivative of the velocity variables. In this dissertation, the DSWE are recast as an evolution equation in time, plus an elliptic constraint equation in space. When discretized (in space), the system of equations takes the form of an (index-1) differential-algebraic equation (DAE). Here the algebraic equation in the DAE captures dispersive effects and consists of a quasi-linear or semi-linear operator. Two strategies are examined to time-step the DSWE in constraint form --- the key novelty is on solving the resulting DAE while avoiding complex nonlinear solutions to the algebraic equations: (i) preconditioned iterative methods are devised to invert the (semi-linear) operator; (ii) semi-implicit time-stepping (ImEx) methods are devised that bypass a full inversion of the quasi/semi-linear operator. Guaranteeing stability for the semi-implicit approach is a nontrivial issue due to the fact that certain stiff terms in the equation are treated explicitly. A stability theory is provided which outlines how to choose the semi-implicit terms in such a way to guarantee numerical (zero) stability.

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