Convergence Analysis of a Krylov Subspace Spectral Method for the 1-D Wave Equation in an Inhomogeneous Medium

Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for partial differential equations (PDEs) that possess stability characteristic of implicit methods. KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE. As a result, KSS methods scale effectively to higher spatial resolution. This paper will present a convergence analysis of a second-order KSS method applied to a 1-D wave equation in an inhomogeneous medium. Numerical experiments that corroborate the established theory are included, along with a discussion of generalizations, such as to higher space dimensions.Comment: 30 pages, 11 figure

Recurrence Relations for Orthogonal Polynomials for PDEs In Polar and Cylindrical Geometrics

This paper introduces two families of orthogonal polynomials on the interval (−1,1), with weight function w(x)=1. The first family satisfies the boundary condition p(1)=0, and the second one satisfies the boundary conditions p(-1)=p(1)=0. These boundary conditions arise naturally from PDEs defined on a disk with Dirichlet boundary conditions and the requirement of regularity in Cartesian coordinates. The families of orthogonal polynomials are obtained by orthogonalizing short linear combinations of Legendre polynomials that satisfy the same boundary conditions. Then, the three-term recurrence relations are derived. Finally, it is shown that from these recurrence relations, one can efficiently compute the corresponding recurrences for generalized Jacobi polynomials that satisfy the same boundary conditions

Modeling of First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods

We solve the first order 2-D reaction–diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical solution, but also an approximate analytical solution when using initial conditions that come from pre-bleach steady states and also general initial conditions, to facilitate asymptotic analysis. Analytical and numerical results are presented. It is observed that although KSS methods are explicit, it is possible to use a time step that is far greater than what the CFL condition would indicate

Numerical Solution of an Extra-wide Angle Parabolic Equation through Diagonalization of a 1-D Indefinite Schr\"{o}dinger Operator with a Piecewise Constant Potential

We present a numerical method for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schr\"{o}dinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made the case of an arbitrary number of pieces. Through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods.Comment: 27 pages, 13 figure

Matrices, Moments and Quadrature: Applications to Time- Dependent Partial Differential Equations

The numerical solution of a time-dependent PDE generally involves the solution of a stiff system of ODEs arising from spatial discretization of the PDE. There are many methods in the literature for solving such systems, such as exponential propagation iterative (EPI) methods, that rely on Krylov projection to compute matrix function-vector products. Unfortunately, as spatial resolution increases, these products require an increasing number of Krylov projection steps, thus drastically increasing computational expense

A New Model for Predicting the Drag and Lift Forces of Turbulent Newtonian Flow on Arbitrarily Shaped Shells on the Seafloor

Currently, all forecasts of currents, waves, and seafloor evolution are limited by a lack of fundamental knowledge and the parameterization of small-scale processes at the seafloor-ocean interface. Commonly used Euler-Lagrange models for sediment transport require parameterizations of the drag and lift forces acting on the particles. However, current parameterizations for these forces only work for spherical particles. In this dissertation we propose a new method for predicting the drag and lift forces on arbitrarily shaped objects at arbitrary orientations with respect to the direction of flow that will ultimately provide models for predicting the sediment sorting processes that lead to the variability of shell fragments on inner shelf seafloors. We wish to develop the drag force parameterization specifically for a limpet shell through the linear regression of force estimated from high-fidelity Reynolds-averaged Navier-Stokes (RANS) simulations in OpenFOAM

Diagonalization of 1-D Differential Operators With Piecewise Constant Coefficients Using the Uncertainty Principle

A highly accurate and efficient numerical method is presented for computing the solution of a 1-D time-dependent partial differential equation in which the spatial differential operator features a piecewise constant coefficient defined on n role= presentation \u3e pieces, in either self-adjoint and non-self-adjoint form, on a finite interval with periodic boundary conditions. The Uncertainty Principle is used to estimate the eigenvalues of the operator. Then, these estimates are used to construct a basis of eigenfunctions for use with a spectral method. The solution is presented as a truncated eigenfunction expansion, where each eigenfunction is a wave function that changes frequencies at the interfaces between different materials. Numerical experiments demonstrate the accuracy, efficiency and scalability of the method in comparison to other methods