Convergence of a Boundary Integral Method for Water Waves

We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269–1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy

Extrapolated regularization of nearly singular integrals on surfaces

We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $\delta$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $\delta$ we can solve for an extrapolated value that has regularization error reduced to $O(\delta^5)$. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$. For convergence as $h \to 0$ we can choose $\delta$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(\delta^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.Comment: submitted to Adv. Comput. Mat

Novel Signal Noise Reduction Method through Cluster Analysis, Applied to Photoplethysmography

Physiological signals can often become contaminated by noise from a variety of origins. In this paper, an algorithm is described for the reduction of sporadic noise from a continuous periodic signal. The design can be used where a sample of a periodic signal is required, for example, when an average pulse is needed for pulse wave analysis and characterization. The algorithm is based on cluster analysis for selecting similar repetitions or pulses from a periodic single. This method selects individual pulses without noise, returns a clean pulse signal, and terminates when a sufficiently clean and representative signal is received. The algorithm is designed to be sufficiently compact to be implemented on a microcontroller embedded within a medical device. It has been validated through the removal of noise from an exemplar photoplethysmography (PPG) signal, showing increasing benefit as the noise contamination of the signal increases. The algorithm design is generalised to be applicable for a wide range of physiological (physical) signals