Bengt Fornberg

A numerical study of steady viscous flow past a circular cylinder

Numerical solutions have been obtained for steady viscous flow past a circular cylinder at Reynolds numbers up to 300. A new technique is proposed for the boundary condition at large distances and an iteration scheme has been developed, based on Newtons method, which circumvents the numerical difficulties previously encountered around and beyond a Reynolds number of 100. Some new trends are observed in the solution shortly before a Reynolds number of 300. As vorticity starts to recirculate back from the end of the wake region, this region becomes wider and shorter. Other flow quantities like position of separation point, drag, pressure and vorticity distributions on the body surface appear to be quite unaffected by this reversal of trends.

A numerical and theoretical study of certain nonlinear wave phenomena

An efficient numerical method is developed for solving nonlinear wave equations typified by the Korteweg-de Vries equation and its generalizations. The method uses a pseudospectral (Fourier transform) treatment of the space dependence together with a leap-frog scheme in time. It is combined with theoretical discussions in the study of a variety of problems including solitary wave interactions, wave breaking, the resolution of initial steps and wells, and the development of nonlinear wavetrain instabilities.

The pseudospectral method; comparisons with finite differences for the elastic wave equation

The pseudospectral (or Fourier) method has been used recently by several investigators for forward seismic modeling. The method is introduced here in two different ways: as a limit of finite differences of increasing orders, and by trigonometric interpolation. An argument based on spectral analysis of a model equation shows that the pseudospectral method (for the accuracies and integration times typical of forward elastic seismic modeling) may require, in each space dimension, as little as a quarter the number of grid points compared to a fourth‐order finite‐difference scheme and one‐sixteenth the number of points as a second‐order finite‐difference scheme. For the total number of points in two dimensions, these factors become 1/16 and 1/256, respectively; in three dimensions, they become 1/64 and 1/4 096, respectively. In a series of test calculations on the two‐dimensional elastic wave equation, only minor degradations are found in cases with variable coefficients and discontinuous interfaces.

A numerical study of some radial basis function based solution methods for elliptic PDEs

Abstract During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ill conditioning, which has previously limited the range that could be used for this parameter. We find that the best values for it often fall outside the range that was previously available. We have also looked at piecewise smooth versus infinitely smooth RBFs, and found that for PDE applications with smooth solutions, the infinitely smooth RBFs are preferable, mainly because they lead to higher accuracy. In a comparison of RBF-based methods against two standard techniques (a second-order finite difference method and a pseudospectral method), the former gave a much superior accuracy.

Interpolation in the Limit of Increasingly Flat Radial Basis Functions

Abstract Many types of radial basis functions, such as multiquadrics, contain a free parameter. In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge. Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials. In the 1-D case, we prove that with simple conditions on the basis function, the interpolants converge to the Lagrange interpolating polynomial. Hence, differentiation of this limit is equivalent to the standard finite difference method. We also summarize some preliminary observations regarding the limit in 2-D.

Classroom Note: Calculation of Weights in Finite Difference Formulas

The classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope (e.g., specialized recursions for centered and staggered approximations, for Adams--Bashforth-, Adams--Moulton-, and BDF-formulas for ODEs, etc.). Two recent algorithms overcome these problems. For equispaced grids, such weights can be found very conveniently with a two-line algorithm when using a symbolic language such as Mathematica (reducing to one line in the case of explicit approximations). For arbitrarily spaced grids, we describe a computationally very inexpensive numerical algorithm.

The Runge phenomenon and spatially variable shape parameters in RBF interpolation

Many studies, mostly empirical, have been devoted to finding an optimal shape parameter for radial basis functions (RBF). When exploring the underlying factors that determine what is a good such choice, we are led to consider the Runge phenomenon (RP; best known in cases of high order polynomial interpolation) as a key error mechanism. This observation suggests that it can be advantageous to let the shape parameter vary spatially, rather than assigning a single value to it. Benefits typically include improvements in both accuracy and numerical conditioning. Still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas. This idea is routinely used when working with splines or finite element methods. However, local refinement with RBFs may cause RP-type errors unless we use a spatially variable shape paremeter. With this enhancement, RBF approximations combine freedom from meshes with spectral accuracy on irregular domains, and furthermore permit local node clustering to improve the resolution wherever this might be needed.

Stable Computations with Gaussian Radial Basis Functions

Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-Pade method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided.

Scattered node compact finite difference-type formulas generated from radial basis functions

In standard equispaced finite difference (FD) formulas, symmetries can make the order of accuracy relatively high compared to the number of nodes in the FD stencil. With scattered nodes, such symmetries are no longer available. The generalization of compact FD formulas that we propose for scattered nodes and radial basis functions (RBFs) achieves the goal of still keeping the number of stencil nodes small without a similar reduction in accuracy. We analyze the accuracy of these new compact RBF-FD formulas by applying them to some model problems, and study the effects of the shape parameter that arises in, for example, the multiquadric radial function.

Steady viscous flow past a circular cylinder up to reynolds number 600

Viscous flow past a circular cylinder becomes unstable around Reynolds number Re = 40. With a numerical technique based on Newtons method and made possible by the use of a supercomputer, steady (but unstable) solutions have been calculated up to Re = 600. It is found that the wake bubble (region with recirculating flow) grows in length approximately linearly with Re. The width increases like Re12 up to Re = 300 at which paint a transition to linear increase with Re begins. At the highest Reynolds numbers we reached, the wake resembles a pair of translating, uniform vortices, both touching the center line. The cylinder; moving in front with the same speed, supplies the vorticity required to balance diffusion.