Tobin A. Driscoll

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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 maps generally require a computer, and although the numerical aspects of these transformations have been studied, there are few software implementations(More)
Spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains, including well-known cases such as the Korteweg–de Vries and nonlinear Schrödinger equations. For the best computational efficiency, one needs also to use high-order methods in time while somehow bypassing the usual severe stability(More)
Abstract. We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initialboundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the(More)
Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given(More)
We propose a new algorithm for computing the Riemann mapping of the unit disk to a polygon, also known as the Schwarz–Christoffel transformation. The new algorithm, CRDT (for cross-ratios of the Delaunay triangulation), is based on cross-ratios of the prevertices, and also on cross-ratios of quadrilaterals in a Delaunay triangulation of the polygon. The(More)
The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor ρ ≤ 1 can be derived by solving a problem of potential theory or conformal mapping. Six(More)
Recently it was proved that there exist nonisometric planar regions that have identical Laplace spectra. That is, one cannot “hear the shape of a drum.” The simplest isospectral regions known are bounded by polygons with reentrant corners. While the isospectrality can be proven mathematically, analytical techniques are unable to produce the eigenvalues(More)
Abstract. We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node(More)
We consider variations of the Adams–Bashforth, backward differentiation, and Runge–Kutta families of time integrators to solve systems of linear wave equations on uniform, time-staggered grids. These methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders. We(More)
A simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part of the model is stable but highly non-normal, so that certain inputs experience a great deal of growth before they eventually decay. The nonlinear terms of the model contribute no energy growth, but recycle some of the linear(More)