Bengt Fornberg

Bengt Fornberg Professor of Applied Mathematics, University of Colorado, Boulder, CO Regular mail: University of Colorado Department of Applied Mathematics, 526 UCB Boulder , CO 80309 USA E-mail: fornberg@colorado.edu Tel: 303-4925915
Faculty member, Center for Research on Training, University of Colorado, Boulder.

Research interests:
My main research interests are in developing, analyzing, and implementing numerical methods, in particular for solving PDEs to high orders of accuracy. Such methods include pseudospectral and high accuracy finite difference methods and, in particular, methods based on radial basis functions (RBFs). The main application areas include computational fluid dynamics, geophysical and astrophysical flows, and different types of wave phenomena.

Some former Ph.D. students

Michelle Ghrist Assoc. Prof. at USAF Academy, Colorado Springs.
Thesis: High-Order Finite Difference Methods for Wave Equations

Grady Wright Assoc. Prof. at Boise State University
Thesis: Radial Basis Function Interpolation: Numerical and Analytical Developments

Julia Zuev Thesis: Recent Advances in Numerical PDEs

Cecile Piret Thesis: Analytical and Numerical Advances in Radial Basis Functions

Jonah Reeger Thesis: A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector methods

Some present collaborators

Natasha Flyer Scientist III at NCAR, Boulder

B.C. Low Senior Scientist at NCAR, Boulder

Elisabeth Larsson Researcher at Uppsala University

André Weideman Prof. at Stellenbosch University

Erik Lehto Post Doc at NCAR, Boulder

CV

Recent papers:

For the actually published articles, please refer directly to the respective journals. In many cases, the links below lead to earlier manuscript versions than those actually published. When the links below lead to the published version, please verify that your use of the article is consistent with relevant copyright regulations.

Painlevé IV: A numerical study of the fundamental domain and beyond (J.A. Reeger and BF), submitted.
Numerical quadrature over the surface of a sphere (BF and J.M. Martel), submitted.
Inverting non-linear dimensionality reduction with scale-free radial basis interpolation (N.D. Monnig, BF and F.G. Meyer), submitted.
Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions (E. Larsson, E. Lehto, A. Heryodono and BF), submitted.
Stability ordinates of Adams predictor-corrector methods (M. Ghrist, J.A. Reeger and BF), submitted.
A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs (C-I.R. Davis and BF), Complex Variables and Elliptic Equations, to appear.
Stable calculation of Gaussian-based RBF-FD stencils (BF, E. Lehto and C. Powell), Computers and Mathematics with Applications, 65 (2013), 627-637.
A computational exploration of the second Painlevé equation (BF and J.A.C. Weideman), Found. Comput. Math., to appear.
Painlevé IV with both parameters zero: A numerical study (J. Reeger and BF), Studies in Applied Math., 130 (2013), 108-133.
Two results concerning the stability of staggered multistep methods (M. Ghrist and BF), SIAM J. Num. Anal.50 (2012), 1849-1860.
A numerical implementation of Fokas boundary integral approach: Laplace's equation on a polygonal domain (BF and N. Flyer), Royal Society Proc. Series A. 467 (2011), 2983-3003.
A numerical methodology for the Painlevé equations (BF and J.A.C. Weideman), Journal of Computational Physics, 230 (2011), 5957-5973.
Radial basis functions: Developments and applications to planetary scale flows (N. Flyer and BF), Computers and Fluids, 46 (2011), 23-32.
Stabilization of RBF-generated finite difference methods for convective PDEs (BF and E. Lehto), Journal of Computational Physics, 230 (2011), 2270-2285.
Stable computations with Gaussian radial basis functions (BF, E. Larsson and N. Flyer), SIAM J. Sci. Comput. 33 (2011), 869-892.
Comparisons between different implementations of Newton's method for a nonlinear Poisson-type PDE (BF, N. Flyer and J.M. Russell), to be submitted.
The Gibbs phenomenon for radial basis functions (BF and Natasha Flyer), in The Gibbs Phenomenon in Various Representations and Applications, ed. A. Jerri, Sampling Publishing, Potsdam, NY, (2011), Chapter 6, 201-224.
Padé-based interpretation and correction of the Gibbs phenomenon (T.A. Driscoll and BF), in The Gibbs Phenomenon in Various Representations and Applications, ed. A. Jerri, Sampling Publishing, Potsdam, NY, (2011), Chapter 5, 173-200.
A finite difference method for free boundary problems, Journal of Computational and Applied Mathematics, 233 (2010), 2831-2840.
Evolution of solitary waves in a two-pycnocline system (M. Nitsche, P.D. Weideman, R. Grimshaw, M. Ghrist and BF), J. Fluid Mech. 642 (2010), 235-277.
Comparisons between pseudospectral and radial basis function derivative approximations (BF, N. Flyer and J.M. Russell), IMA Journal of Numerical Analysis, 30 (2010), 149-172.
Magnetic relaxation in the solar corona (K. Miller, BF, N. Flyer and B.C. Low), The Astrophysical Journal, 690 (2009), 720-733.
Steady axisymmetric vortex flows with swirl and shear (A. Elcrat, BF and K. Miller), J. Fluid Mech, 613 (2008), 395-410.
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere (BF and C. Piret), Journal of Computational Physics, 227 (2008), 2758-2780.
Locality properties of radial basis function expansion coefficients for equispaced interpolation (BF, Natasha Flyer, S. Hovde and C. Piret), IMA Journal of Numerical Analysis, 28 (2008), 121-142.
A stable algorithm for flat radial basis functions on a sphere (BF and C. Piret), SIAM J. Sci. Comp. 30 (2007), 60-80.
The Runge phenomenon and spatially variable shape parameters in RBF interpolation (BF and J. Zuev), Computers and Mathematics with Applications, 54 (2007), 379-398.
Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equations (BF, J. Zuev and J. Lee), Journal of Computational and Applied Mathematics, 200 (2007), 178-192.
A pseudospectral fictitious point method for high order initial-boundary value problems , SIAM J. Sci. Comp. 28 (2006), 1716-1729.
A new class of oscillatory radial basis functions (BF, E. Larsson and G. Wright), Computers and Mathematics with Applications 51 (2006), 1209-1222.
Scattered node compact finite difference-type formulas generated from radial basis functions (G. Wright and BF), Journal of Computational Physics 212 (2006), 99-123.
Stability of vortices in equilibrium with a cylinder (A. Elcrat, BF, and K. Miller), J. Fluid. Mech. 544 (2005), 53-68.
Magnetic-field confinement in the solar corona. II. Field-plasma interaction (N. Flyer, BF, S. Thomas and B.C. Low), The Astrophysical Journal 631 (2005), 1239-1259.
Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids (BF and N. Flyer). Advances in Computational Mathematics 23 (2005), 5-20.
Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions (E. Larsson and BF). Computers and Mathematics with Applications 49 (2005), 103-130.
Magnetic-field confinement in the solar corona. I. Force-free magnetic fields (N. Flyer, BF, S. Thomas and B.C. Low). The Astrophysical Journal 606 (2004), 1210-1222.
Stable computation of multiquadric interpolants for all values of the shape parameter (BF and G. Wright). Computers and Mathematics with Applications 48 (2004), 853-867.
Some unconditionally stable time stepping methods for the 3-D Maxwell's equations (J. Lee and BF). Journal of Computational and Applied Mathematics 166 (2004), 497-523.
Some observations regarding interpolants in the limit of flat radial basis functions (BF, G. Wright and E. Larsson ), Computers and Mathematics with Applications 47 (2004), 37-55.
On the nature of initial-boundary value solutions for dispersive equations (N. Flyer and BF). SIAM J. Appl. Math.64 (2003), 546-564.
A numerical study of some radial basis function based solution methods for elliptic PDEs (E. Larsson and BF), Computers and Mathematics with Applications, 46 (2003), 891-902.
Accurate numerical resolution of transients in initial-boundary value problems for the heat equation (N. Flyer and BF), Journal of Computational Physics 184/2 (2003), 526-539.
Some numerical techniques for Maxwell's equations in different types of geometries , Topics in Computational Wave Propagation, Lecture notes in Computational Science and Engineering 31, Springer Verlag (2003), 265-299.
A split step approach for the 3-D Maxwell's equations (J. Lee and BF), Journal of Computational and Applied Mathematics 158 (2003), 485-505.
A short proof of the unconditional stability of the ADI-FDTD scheme , CU APPM Preprint #472.
Interpolation in the limit of increasingly flat radial basis functions (T.A. Driscoll and BF), Computers and Mathematics with Applications, 43(2002), 413-422.
Observations on the behavior of radial basis functions near boundaries (BF, T.A. Driscoll, G. Wright and R. Charles), Computers and Mathematics with Applications 43(2002), 473-490.
Some steady axisymmetric vortex flows past a sphere (A. Elcrat, BF and K. Miller), J. Fluid Mech. 433 (2001), 315-328.
A Padé-based algorithm for overcoming the Gibbs' phenomenon (T.A. Driscoll and BF), Numerical Algorithms 26 (2001), 77-92.
Note on nonsymmetric finite differences for Maxwell's equations (T.A. Driscoll and BF), J. Comput. Phys.161 (2000), 723-727.
Staggered time integrators for wave equations (BF, M. Ghrist and T.A. Driscoll), SIAM J. Num. Anal. 38 (2000), 718-741.
Some steady vortex flows past a circular cylinder (A. Elcrat, BF, M. Horn and K. Miller), J. Fluid. Mech. 409 (2000), 13-27.
A fast spectral algorithm for nonlinear wave equations with linear dispersion (BF and T.A. Driscoll), J. Comput. Phys. 155 (1999), 456-467.
Spatial finite difference approximations for wave-type equations (BF and M. Ghrist). SIAM J. Num. Anal. 37 (1999), 105-130.
Block pseudospectral methods for Maxwell's equations: II. Two-dimensional, discontinuous-coefficient case (T.A. Driscoll and BF). SIAM J. Sci. Comput. 21 (1999), 1146-1167.
On the chance of freak waves at sea (B.S. White and BF). J. Fluid. Mech. 355 (1998), 113-138.
A block pseudospectral method for Maxwell's equations: I. One-dimensional case (T.A. Driscoll and BF). J. Comput. Phys. 140 (1998), 47-65.
Calculation of weights in finite difference formulas. SIAM Review 40 Nr 3, (1998), 685-691.
Comparison of finite difference- and pseudospectral methods for convective flow over a sphere (BF and D. Merrill).Geophys. Res. Lett. 24, No 24 (1997), 3245-3248. (Summarizes Dave Merrill's Master's Thesis).

A few less recent papers:

A compact fourth order finite difference scheme for the steady incompressible Navier-Stokes equations , (M. Li, T. Tang and BF), Int. J. for Numerical Methods in Fluids, 20 (1995), 1137-1151.
Computing steady incompressible flows past blunt bodies - A historical overview , in Numerical Methods for Fluid Dynamics IV (Ed. M.J. Baines and K.W. Morton) Oxford Univ. Press (1993), 115-134.
Steady incompressible flow past a row of circular cylinders , J. Fluid Mech. 225 (1991), 655-671.
Generation of finite difference formulas on arbitrarily spaced grids , Mathematics of Computation, 51 (1988), 699-706.
Steady viscous flow past a sphere at high Reynolds numbers , J. Fluid Mech. 190 (1988), 471-489.
The pseudospectral method: Accurate representation of interfaces in elastic wave calculations, Geophysics, 53 (1988), 625-637.
The pseudospectral method: Comparisons with finite differences for the elastic wave equation, Geophysics, 52 (1987), 483-501.
Steady viscous flow past a circular cylinder up to Reynolds number 600, Journal of Computational Physics, 61 (1985), 297-320.
Algorithm 579, CPSC: Complex Power Series Coefficients, ACM Transactions in Mathematical Software, 7 (1981), 542-547.
Numerical differentiation of analytic functions, ACM Transactions in Mathematical Software, 7 (1981), 512-526.
A numerical study of steady viscous flow past a circular cylinder , J. Fluid Mech. 98 (1980), 819-855.
A numerical method for conformal mappings , SIAM J. Sci. Stat. Comp., 1(1980), 386-400.
A numerical and theoretical study of certain nonlinear wave phenomena (BF and G.B. Whitham), Phil. Trans. Royal Soc. London, Ser A, Vol 289 (1978), 373-404.
A numerical study of 2-D turbulence , Journal of Computational Physics, 25 (1977), 1-31.
On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation , Mathematics of Computation, 27 (1973), 45-57.

SOME OTHER ITEMS OF INTEREST

University of Colorado at Boulder
C.U.Boulder Department of Applied Mathematics
Numerical Analysis Network
History of Mathematics
Scholarpedia article: Finite difference method

Michelle Ghrist	Assoc. Prof. at USAF Academy, Colorado Springs. Thesis: High-Order Finite Difference Methods for Wave Equations
Grady Wright	Assoc. Prof. at Boise State University Thesis: Radial Basis Function Interpolation: Numerical and Analytical Developments
Julia Zuev	Thesis: Recent Advances in Numerical PDEs
Cecile Piret	Thesis: Analytical and Numerical Advances in Radial Basis Functions
Jonah Reeger	Thesis: A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector methods

Natasha Flyer	Scientist III at NCAR, Boulder
B.C. Low	Senior Scientist at NCAR, Boulder
Elisabeth Larsson	Researcher at Uppsala University
André Weideman	Prof. at Stellenbosch University
Erik Lehto	Post Doc at NCAR, Boulder