arXiv subjects

James V. Lambers

Publications and source records attributed to James V. Lambers.

4 recordsLinked to original sources

arXiv2023

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

This paper presents a convergence analysis of a Krylov subspace spectral (KSS) method applied to a 1-D wave equation in an inhomogeneous medium. It will be shown that for sufficiently regular initial data, this KSS method yields unconditional stability, spectral accuracy in space, and second-order accuracy in time, in the case of constant wave speed and a bandlimited reaction term coefficient. Numerical experiments that corroborate the established theory are included, along with an investigation of generalizations, such as to higher space dimensions and nonlinear PDEs, that features performance comparisons with other Krylov subspace-based time-stepping methods. This paper also includes the first stability analysis of a KSS method that does not assume a bandlimited reaction term coefficient.

math.NA↗

arXiv2023

Acoustic singular surfaces in an exponential class of inhomogeneous gases: A new numerical approach based on Krylov subspace spectral methodologies

We investigate the propagation of acoustic singular surfaces, specifically, linear shock waves and nonlinear acceleration waves, in a class of inhomogeneous gases whose ambient mass density varies exponentially. Employing the mathematical tools of singular surface theory, we first determine the evolution of both the jump amplitudes and the locations/velocities of their associated wave-fronts, along with a variety of related analytical results. We then turn to what have become known as Krylov subspace spectral (KSS) methods to numerically simulate the evolution of the full waveforms under consideration. These are not only performed quite efficiently, since KSS allows the use of `large' CFL numbers, but also quite accurately, in the sense of capturing theoretically-predicted features of the solution profiles more faithfully than other time-stepping methods, since KSS customizes the computation of the components of the solution corresponding to the different frequencies involved. The presentation concludes with a listing of possible, acoustics-related, follow-on studies.

math.NA↗

arXiv2022

Numerical Solution of an Extra-wide Angle Parabolic Equation through Diagonalization of a 1-D Indefinite Schrödinger 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 Schrödinger 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.

math.NA↗

arXiv2015

Approximation of the Scattering Amplitude using Nonsymmetric Saddle Point Matrices

In this paper we examine iterative methods for solving the forward ($A{\bf x}={\bf b}$) and adjoint ($A^{T}{\bf y}={\bf g}$) systems of linear equations used to approximate the scattering amplitude, defined by ${\bf g}^{T}{\bf x}={\bf y}^{T}{\bf b}$. Based on an idea first proposed by Gene Golub, we use a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed so as to have a real positive spectrum. Numerical experiments show that this method is more consistent than known methods for computing the scattering amplitude such as GLSQR or QMR. We then demonstrate that when combined with known preconditioning techniques, the proposed method exhibits more rapid convergence than state-of-the-art iterative methods for nonsymmetric systems.

math.NA↗