This dissertation consists of two separate chapters. In the first chapter, we present an algorithm for computing a Smith normal form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column) to obtain the Smith normal form. The performance of the algorithm in exact arithmetic is reported for several test cases.

The second chapter is devoted to a numerical method for computing nontrivial time-periodic, gravity-driven water waves with or without surface tension. This method is essentially a shooting method formulated as a minimization problem. The objective function depends on the initial conditions and the proposed period, and measures deviation from time-periodicity. We adapt an adjoint-based optimal control method to rapidly compute the gradient of the functional. The main technical challenge involves handling the nonlocal Dirichlet to Neumann operator of the water wave equations in the adjoint formulation. Several families of traveling waves and symmetric breathers are simulated. In the latter case, we observe disconnections in the bifurcation curves due to nonlinear resonances at critical bifurcation parameters.

Axisymmetric flow around a bubble rising in a 3D fluid has been studied intensively both numerically and experimentally over the years, but in the high Reynolds number regime, no matter for small oblate or elongated bubbles, current mathematical models face problems such as introducing an unnecessary singularity at the stagnation point, missing proper lower boundary conditions, and solving with low order finite difference methods, which leads to large numerical errors. Modeling the problem by both potential flow and viscous potential flow, we can represent the velocity potential using layer potentials and compute the bubble shape with spectral accuracy, with average numerical errors around $10^{-9}$. A dimensionless inviscid model is first built to study steady flow around a bubble in an infinitely long tube in the case of zero gravity. With a comprehensive discussion on parametrization, singular integrals, and numerical quadrature, we solve both oblate and elongated bubble shapes accurately and find different solution branches of bubble shapes characterized by the number of humps. These solution branches relating the Weber number and cross-section arc-length suggest that for any nonzero surface tension, there exists a countably infinite number of solutions. When there is gravity, kinematic viscosity is introduced to balance out the normal stress on the bubble surface. Different shapes of steadily rising bubbles under different Froude numbers are presented.

We also study the corresponding time-dependent problem to illustrate the dynamics of unsteady bubbles, both small and elongated, and to confirm the non-existence of steadily rising bubbles in the inviscid model with nonzero gravity. The approximation of dipole density by spherical harmonics is computed to remove the hyper-singularity in the normal derivatives of the double layer potential on the bubble surface.

Due to the importance of accurate evaluation of orthogonal polynomials in the boundary integral method, we introduce a new way to evaluate orthogonal polynomials more accurately near the endpoints of the integration interval by evaluating a newly created set of associated orthogonal polynomials at corresponding points. Various ways to evaluate the associated orthogonal polynomials are discussed and implemented. Among them, the best method can achieve round-off error accuracy even for end-point evaluation of generic high-degree Jacobi polynomials and generalized Laguerre polynomials, which is 3 digits more accurate than the classic recurrence method in double precision when the degree is around 400. Based on the accurate evaluation, we perform one iteration of Newton's method to achieve more accurate quadrature abscissas near endpoints. We also calculate quadrature weights using the Christoffel formula based on our accurate evaluation. The resulting errors in abscissas and weights are compared to those of the Golub-Welsch algorithm, with our method being more accurate. Strategies on generic weight functions are given and tested on Maxwell polynomials of current interest in plasma physics.

In this thesis, we study the problem of two dimensional spatially quasi-periodic gravity-capillary waves on the surface of an ideal fluid of infinite depth. We formulate the water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of both the initial value problem and the traveling wave problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one dimensional quasi-periodic functions are represented by two dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two time-stepping schemes of the initial value problem are proposed, an explicit Runge-Kutta (ERK) method and an exponential time-differencing (ETD) scheme. We present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. We formulate the traveling wave problem as a nonlinear least squares problem that we solve using a variant of the Levenberg-Marquardt method. Two types of quasi-periodic traveling solutions are computed: small-amplitude solutions that bifurcate from the zero solution and large-amplitude solutions that bifurcate from finite-amplitude periodic traveling solutions. Solutions of the first type are identified by two bifurcation parameters. We also compute the leading terms of the asymptotic expansion of the solution using these parameters. For solutions of the second type, we apply the Fourier-Bloch decomposition to study the linearization around periodic traveling solutions and obtain a one-parameter family of quasi-periodic solutions bifurcating from the branch of periodic solutions. As an example, we compute a branch of quasi-periodic overturning traveling solutions that bifurcate from a periodic overturning traveling solution.

In 1831, Michael Faraday observed the formation of standing waves on the surface of a vibrating fluid body. Subsequent experiments have revealed the existence of a rich tapestry of patterned states that can be accessed by varying the frequency and amplitude of the vibration and have spurred vast research in hydrodynamics and pattern formation. These include linear analyses to determine the conditions for the onset of the patterns, weakly nonlinear studies to understand pattern selection, and dynamical systems approaches to study mode competition and chaos. Recently, there has been some work towards numerical simulations in various three-dimensional geometries. These methods however possess low orders of accuracy, making them unsuitable for nonlinear regimes.

We present a new technique for fast and accurate simulations of nonlinear Faraday waves in a cylinder. Beginning from a viscous potential flow model, we generalize the Transformed Field Expansion to this geometry for finding the highly non-local Dirichlet-to-Neumann operator (DNO) for the Laplace equation. A spectral method relying on Zernike polynomials is developed to rapidly compute the bulk potential. We prove the effectiveness of representing functions on the unit disc in terms of these polynomials and also show that the DNO algorithm possesses spectral accuracy, unlike a method based on Bessel functions.

The free surface evolution equations are solved in time using Picard iterations carried out by left-Radau quadrature. The results are in perfect agreement with the instability thresholds and surface patterns predicted for the linearized problem. The nonlinear simulations reproduce several qualitative features observed experimentally. In addition, by enabling one to switch between various nonlinear regimes, the technique allows a precise determination of the mechanisms triggering various experimental observations.