Numerical Study of the Helmholtz Equation with Large Wavenumbers
Creators: Michelle, Michelle
Publisher: name: University of Alberta Library
Year: 2022

<p>The Helmholtz equation is a fundamental wave propagation model in the time-harmonic setting, which appears in many applications such as electromagnetics, geophysics, and ocean acoustics. It is challenging and computationally expensive to solve due to (1) its highly oscillating solution and (2) the large ill-conditioned/unstable sign-indefinite linear system arising from standard discretizations, especially when a large wavenumber is present. In this thesis, we develop and extensively study high order compact finite difference methods (FDMs) and a wavelet Galerkin method for the Helmholtz equation in various settings.</p>

<p>In Chapter 1, we provide some background on the Helmholtz equation and wavelets. </p>

<p>In Chapter 2, we introduce the Dirac Assisted Tree (DAT) method coupled with an arbitrarily high order compact 1D FDM. DAT successfully overcomes the massive ill-conditioned linear system associated with the Helmholtz equation by breaking a global problem into small much better conditioned linking and local problems, as well as harnessing parallel computing resources. DAT is effective in solving 1D heterogeneous and special 2D Helmholtz equations with arbitrarily large wavenumbers. </p>

<p>In Chapter 3, we propose a new pollution minimizing sixth order compact FDM for the 2D Helmholtz equation with interfaces and mixed boundary conditions. The new pollution minimization strategy we employ is based on the average truncation error of plane waves. Compared to existing FDMs, the errors of our method are several orders of magnitude lower. </p>

<p>In Chapter 4, we present new sharp wavenumber-explicit stability bounds for the 2D Helmholtz equation with mixed inhomogeneous boundary conditions. Such bounds are crucial in the analysis and development of numerical schemes, since they describe how the solution behaves for given data. Establishing these bounds is difficult, since they highly depend on boundary conditions and the domain’s geometry. These findings motivate a future development of a numerical method, which uses our stability bounds to strategically select dominant Fourier coefficients in the solution. </p>

<p>Wavelets are sparse multiscale representation systems built from refinable functions (i.e., functions that can be expressed as dilated and shifted versions of themselves; e.g., B-splines and Hermite splines). A Riesz wavelet on a bounded domain in $\R^d$ (e.g., $[0,1]^d$) is obtained from the tensor product of 1D Riesz wavelets on a bounded interval. Hence, the efficacy of a wavelet method in solving multidimensional problems (e.g., image processing and numerical PDEs) relies on the optimal construction of a wavelet basis on an interval. Many available constructions suffer from shortcomings. For example, some boundary elements may have reduced vanishing moments, which adversely impact the overall sparsity of the system. Furthermore, all existing constructions in the literature are applicable only to particular examples or a very narrow family of wavelet bases. A natural question is whether a systematic construction procedure for any compactly supported wavelet basis exists. In Chapter 5, we fully answer this long-standing problem in wavelet analysis. We propose and study two systematic approaches that construct all locally supported biorthogonal multiwavelets on an interval from any compactly supported biorthogonal multiwavelets on $\R$. </p>

<p>In Chapter 6, we apply the direct approach in the previous chapter to construct 1D Riesz wavelets on the unit interval, and subsequently a 2D Riesz wavelet on the unit square via tensor product. The latter is used in the Galerkin scheme to solve the 2D Helmholtz equation with a non-local boundary condition, which models electromagnetic scattering from a large cavity. The implementation of our method is very efficient. Also, our numerical experiments indicate that the coefficient matrix of our wavelet Galerkin method is much better conditioned (i.e., much more stable) than that of a standard Galerkin method.</p>