Fluid flows that are smooth at low speeds become unstable and then turbulent higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow testing for eigenvalues linearized problem, but results such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role hydrodynamic instability. A reconciliation these findings traditional analysis is presented based on "pseudospectra" which imply small...

The Schwarz-Christoffel transformation and its variations yield formulas for conformal maps from standard regions to the interiors or exteriors of possibly unbounded polygons. Computations involving these generally require a computer, although numerical aspects transformations have been studied, there are few software implementations that widely available suited general use. Toolbox MATLAB is new implementation disk, half-plane, strip, rectangle domains polygon interiors, disk exteriors....

A simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part the stable but highly non-normal, so that certain inputs experience great deal growth before they eventually decay. nonlinear terms contribute no energy growth, recycle some outputs into inputs, closing feedback loop and allowing initially small solutions ‘‘bootstrap’’ much larger amplitude. Although different choices parameters nonlinearity lead variety...

The theory of the convergence Krylov subspace iterations for linear systems equations (conjugate gradients, biconjugate GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation this kind, an estimated asymptotic factor $\rho \le 1$ can be derived by solving problem potential or conformal mapping. Six approximations are involved in relating actual to scalar estimate. These six discussed asystematic way illustrated sequence examples computed with tools numerical mapping semidefinite...

Boundary conditions in spectral collocation methods are typically imposed by removing some rows of the discretized differential operator and replacing them with others that enforce required at boundary. A new approach based upon resampling differentiated polynomials into a lower-degree subspace makes differentiation matrices, operators built from them, rectangular without any row deletions. Then, boundary interface can be adjoined to yield square system. The resulting method is both flexible...

We consider variations of the Adams--Bashforth, backward differentiation, and Runge--Kutta families time integrators to solve systems linear wave equations on uniform, time-staggered grids. These methods are found have smaller local truncation errors allow larger stable steps than traditional nonstaggered versions equivalent orders. investigate accuracy stability these analytically, experimentally, through use a novel root portrait technique.

We propose a new algorithm for computing the Riemann mapping of unit disk to polygon, also known as Schwarz--Christoffel transformation. The algorithm, CRDT (for cross-ratios Delaunay triangulation), is based on prevertices, and quadrilaterals in triangulation polygon. produces an accurate representation even presence arbitrary long, thin regions unlike any previous conformal algorithm. believe that solves all difficulties with crowding global convergence, although these facts depend...

Recently it was proved that there exist nonisometric planar regions have identical Laplace spectra. That is, one cannot "hear the shape of a drum." The simplest isospectral known are bounded by polygons with reentrant corners. While isospectrality can be proven mathematically, analytical techniques unable to produce eigenvalues themselves. Furthermore, standard numerical methods for computing eigenvalues, such as adaptive finite elements, highly inefficient. Physical experiments been...

We consider model problems for the tear film over multiple blink cycles that utilize a single equation film; non-linear partial differential governs thickness arises from lubrication theory. The two models we arise considering absence of naturally occurring surfactant and case when is strongly affecting surface tension. considered on time-varying domain length with specified volume flux at each end; only one end moving, which analogous to upper eyelid moving blink. Realistic lid motion...

The Schwarz--Christoffel Toolbox (SC Toolbox) for MATLAB, first released in 1994, made possible the interactive creation and visualization of conformal maps to regions bounded by polygons. most recent release supports new features, including an object-oriented command-line interface model, algorithms multiply elongated multiple-sheeted regions, a module solving Laplace's equation on polygon with Dirichlet homogeneous Neumann conditions. Brief examples are given demonstrate capabilities.

We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these are susceptible to the Runge phenomenon, not only in limit increasingly flat functions, but also finite shape parameter case. show there exist interpolation node distributions prevent such phenomena allow stable approximations. polynomials provides an explicitinterpolation formula avoids difficulties inverting matrices, while imposing restrictions on...

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to canonical radial or circular slit domain, domains with both and boundary components are derived implemented numerically graphically. The generated by analytic continuation the reflection principle. Convergence of infinite products is proved sufficiently well-separated components. Some recent progress in numerical implementation presented.