Research and Publications

Books

• B. Fornberg and C. Piret, Complex Variables and Analytic Functions: An Illustrated Introduction (SIAM, 2020) 
• B. Fornberg and N. Flyer: A Primer on Radial Basis Functions with Applications to the Geosciences (SIAM, 2015)
• B. Fornberg: A Practical Guide to Pseudospectral Methods (Cambridge University Press, 1996)

Select presentations

• Quadrature formulas, the complex plane, fractional derivatives and undergrond sensing (Air Force Institute of Technology, 05/23/24)
• Some new perpectives on finite differences and the trapezoidal rule - Along the real axis and in the complex plane (Florida State University, 01/16/24)
• Finite difference formulas and numerical contour integration in the complex plane (INI, Cambridge, England, 07/25/23, and UC Colorado Springs 10/12/23) 
• Computing Fractional Derivatives of Analytic Functions (CU Boulder 10/06/22)
• Finite difference formulas in the complex plane (CU Boulder, 11/3/21)
• Enhancing the trapezoidal rule in the complex plane and along the real axis (UC Irvine, 10/26/20)
• Euler-Maclaurin without Analytic Derivatives (CU Boulder, 02/07/20)
• Improving the Accuracy of the Trapezoidal Rule (Univ. of New Mexico, 01/24/19) 
• Radial Basis Function Generated Finite Differences (RBF-FD): New Computational Opportunities for Solving PDEs (Oxford University, 8/25/15).
• Numerical Computations on the Painlevé Equations (University College, London, 04/21/15).
• Numerical Quadrature over the Surface of a Sphere (Michigan Tech, 03/26/15).

Select papers  (in most cases, preprint versions of the actual published version) 

• High-order numerical method for solving elliptic partial differential equations on unfitted node sets (M.E. Nielsen and BF), arXiv: 2407.15825   (2024). 
• Analytic continuation: A tool for areomagnetic data interpretation (J.B. Thurston and BF), The Leading Edge, April 2024, 154-160.  
• Node subsampling for multilevel meshfree elliptic PDE solvers (A.P. Lawrence, M.E. Nielsen and BF), Comp. Math. Applic. 164 (2024), 79-94.
• Enhanced trapezoidal rule for discontinuous functions (BF and A. Lawrence),  Journal of Computational Physics, 491 (2023) Article nr: 112386..
• Computation of fractional derivatives of analytic functions (BF and C. Piret), J. Sci. Comput., 96:79 (2023).
• Numerical computation of fractional derivatives of Caputo type (BF, C. Piret and A. Higgins), Fundamentals of Fractional Calculus, D.K. Singh and M. Yavuz (eds.), CRC Press, to appear.
• Accelerating explicit time-stepping with spatially variable time steps through machine learning (K. van der Sande, N. Flyer and BF), J. Sci. Comput., 96:31 (2023).
• Infinite order accuracy limit of finite difference formulas in the complex plane, IMA J. Numerical Analysis, 43 (2023), 3055-3072.
• Fully numerical Laplace transform methods (J.A.C. Weideman and BF), Numerical Algorithms, 92 (2023), 985-1006.
• Finite difference formulas in the complex plane, Numerical Algorithms, 90 (2022), 1305-1326.
• On the infinite order limit of Hermite-based finite difference schemes (D. Abrahansen and BF), SIAM J. Numerical Analysis, 59 (2021), 1857-1874.
• A parallel-in-time approach for wave-type PDEs (A.C. Ellison and BF), Numerische Mathematik 148 (1) (2021), 79-98.
• Solving the Korteweg-de Vries equation with Hermite-based finite differences (D. Abrahansen and BF), Appl. Math. and Computation, 401 (2021), Article Nr 126101.
• Fast variable density 3-D node generation, (K. van der Sande and BF), SIAM J. Scientific Computing 43 (1) (2021) A242-A257.
• Euler-Maclaurin expansions without analytic derivatives, Proc. Royal Soc. London, A, Vol 476, Article Nr 20200441 (2020). 
• Generalizing the trapezoidal rule in the complex plane, Numerical Algorithms 87 (2021), 187-202.
• Contour integrals of analytic functions given on a grid in the complex plane, IMA J. Numerical Analysis, 41 (2021), 814-825.
• Improving the accuracy of the trapezoidal rule, SIAM Review, Education Section 63 (1) (2021), 167-180.
• An algorithm for calculating Hermite-based finite difference weights, IMA J. Numerical Analysis,  41 (2021), 801-813.
• Transport schemes in spherical geometries using spline-based RBF-FD with polynomials (D. Gunderman, N. Flyer and BF), Journal of Computational Physics 408 (2020), Article Nr: 109256.
• The radial basis functions method for improved numerical approximations of geological processes in heterogeneous systems (C. Piret, N. Dissanayake, J. Gierke and BF), Mathematical Geosciences, 52 (2020), 477-497.
• Explicit time stepping of PDEs with local refinement in space-time (D. Abrahamsen and BF), Journal of Scientific Computing, 81 (2019), 1945-1962.
• On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries (V. Bayona, N. Flyer and BF), Journal of Computational Physics, 380 (2019), 378-399.  
• An improved Gregory-like method for 1-D quadrature (BF and J.A. Reeger), Numerische Mathematik, 141 (2019), 1-19. 
• A computational exploration of the McCoy-Tracy-Wu solutions of the third Painlevé equation (M. Fasondini, BF and J.A.C. Weideman), Physica D, 363 (2018), 18-43.
• On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains (M.J. Colbrook, N. Flyer and BF), Journal of Computational Physics, 374 (2018), 996-1016.   
• Numerical quadrature over smooth surfaces with boundaries (J.A. Reeger and BF), Journal of Computational Physics, 355 (2018), 176-190.  
• Using radial basis function-generated finite differences (RBF-FD) to solve heat transfer equilibrium problems in domains with interfaces (B. Martin and BF), Eng. Anal. w. Boundary Elements, 79 (2017), 38-48. 
• Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces (M. Fasondini, BF, J.A.C. Weideman),  Journal of Computational Physics, 344 (2017), 36-50. 
• Seismic modeling with radial basis function-generated finite differences (RBF-FD) - a simplified treatment of interfaces (B. Martin and BF), Journal of Computational Physics, 335 (2017), 828-845. 
• On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs (V. Bayona, N. Flyer, BF and G.A. Barnett),  Journal of Computational Physics, 332 (2017), 257-273.  
• Stable computations with flat radial basis functions using vector-valued rational approximations (G.B. Wright and BF), Journal of Computational Physics, 331 (2017), 137-156.  
• Fast calculation of Laurent expansions for matrix inverses, Journal of Computational Physics, 326 (2016), 722-732.
• On the role of  polynomials in RBF-FD approximations: I. Interpolation and accuracy (N. Flyer, BF, G.A. Barnett and V. Bayona), Journal of Computational Physics, 321 (2016), 21-38.
• Numerical quadrature over the surface of a sphere (J.A. Reeger and BF), Studies in Applied Math. 137 (2015), 174-188.
• Solving PDEs with radial basis functions (BF and N. Flyer), Acta Numerica 24 (2015), 215-258.
• A computational overview of the solution space of the imaginary Painlevé II equation (BF and J.A.C. Weideman), Physica D, 309 (2015), 108-118.
• Seismic modeling with radial basis function-generated finite differences (RBF-FD) (B. Martin, BF and A. St-Cyr), Geophysics, 80, No 4 (2015), T137-T146.
• Fast generation of 2-D node distributions for mesh-free PDE discretizations (BF and N. Flyer), Comp. Math. Applic. 69 (2015), 531-544. (Files to run the main test problem: main_script.m, node_placing.m, radius_trui.m, trui.png.)
• Stability ordinates of Adams predictor-corrector methods (M. Ghrist, BF and J.A. Reeger), BIT, 55 (2015), 733-750.
• Painlevé IV: A numerical study of the fundamental domain and beyond (J.A. Reeger and BF),  Physica D 280-281 (2014), 1-13.
• Radial basis function-generated finite differences: A mesh-free method for computational geosciences  (N. Flyer, G.B. Wright and BF), Handbook of Geomathematics, 2014, Springer.
• A computational exploration of the second Painlevé equation (BF and J.A.C. Weideman), Found. Comput. Math., 14 (2014), 985-1016.
• Some observations regarding steady laminar flows past bluff bodies (BF and A. Elcrat), Phil. Trans. Roy. Soc. London, Ser. A, 372 (2014), Article 20130353.
• On spherical harmonics based numerical quadrature over the surface of a sphere (BF and J.M. Martel), Adv. Comp. Math., 40 (2014), 1169-1184.
• Inverting non-linear dimensionality reduction with scale-free radial basis interpolation (N.D. Monnig, BF and F.G. Meyer), Appl. Comput. Harm. Anal., 37 (2014), 162-170.
• 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, 59 (2014), 564-577.
• Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions (E. Larsson, E. Lehto, A. Heryodono and BF), SIAM J. Sci. Comp. 35 (2013), A2096-A2119.
• Stable calculation of Gaussian-based RBF-FD stencils (BF, E. Lehto and C. Powell), Comp. Math. Applic., 65 (2013), 627-637.
• 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), Comp. Math. Applic., 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 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.
• A method for acceleration of the convergence of infinite series , (A. Beckman, BF and A. Tengvald), BIT, 9 (1969), 78-80.