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Books

• B. Fornberg and C. Piret, Complex Variables and Analytic Functions: An Illustrated Introduction Ìý
• B. Fornberg and N. Flyer: A Primer on Radial Basis Functions with Applications to the GeosciencesÌý
• B. Fornberg: A Practical Guide to Pseudospectral MethodsÌý

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 planeÌý(Florida State University, 01/16/24)
• Finite difference formulas and numerical contour integration in the complex planeÌý(INI, Cambridge, England, 07/25/23, and UC Colorado Springs 10/12/23)Ìý
• Computing Fractional Derivatives of Analytic Functions (CU Â鶹ӰԺ 10/06/22)
• Finite difference formulas in the complex plane (CU Â鶹ӰԺ, 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 DerivativesÌý(CU Â鶹ӰԺ, 02/07/20)
• Improving the Accuracy of the Trapezoidal Rule (Univ. of New Mexico, 01/24/19)Ìý
• Radial Basis Function Generated Finite Differences (RBF-FD): New Computational Opportunities for Solving PDEsÌý(Oxford University, 8/25/15).
• Numerical Computations on the Painlevé EquationsÌý(University College, London, 04/21/15).
• Numerical Quadrature over the Surface of a SphereÌý(Michigan Tech, 03/26/15).

Select papersÌý (in most cases, preprint versions of theÌýactual published version)Ìý

• High-order numerical method for solving elliptic partial differential equations on unfitted node sets (M.E. Nielsen and BF), arXiv: 2407.15825Ìý Ìý(2024).Ìý
• Analytic continuation: A tool for areomagnetic data interpretation (J.B. Thurston and BF), The Leading Edge, April 2024, 154-160.ÌýÌý
• Node subsampling for multilevel meshfree elliptic PDE solvers (A.P. Lawrence, M.E. NielsenÌýand BF), Comp. Math. Applic. 164 (2024), 79-94.
• Enhanced trapezoidalÌýrule for discontinuous functions (BF and A. Lawrence),ÌýÌýJournal of Computational Physics, 491 (2023) Article nr: 112386..
• Computation of fractional derivatives of analytic functionsÌý(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,ÌýIMA J. Numerical Analysis, 43 (2023), 3055-3072.
• Fully numerical Laplace transform methodsÌý(J.A.C. Weideman and BF), Numerical Algorithms, 92 (2023), 985-1006.
• , 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 MathematikÌý148 (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 ComputingÌý43 (1) (2021) A242-A257.
• Euler-Maclaurin expansions without analytic derivatives, Proc. Royal Soc.ÌýLondon, A, Vol 476, Article Nr 20200441Ìý(2020).Ìý
• Generalizing the trapezoidal rule in the complex plane,ÌýNumerical AlgorithmsÌý87 (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,ÌýSIAM Review, Education SectionÌý63 (1) (2021), 167-180.
• An algorithm for calculating Hermite-based finite difference weights,ÌýIMA J. Numerical Analysis,ÌýÌý41 (2021), 801-813.
• Transport schemes in spherical geometries using spline-based RBF-FD with polynomialsÌý(D. Gunderman, N. Flyer and BF), Journal of Computational Physics 408 (2020), Article Nr: 109256.
• (C. Piret, N. Dissanayake, J. Gierke and BF),ÌýMathematical 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),ÌýJournal of Computational Physics, 380 (2019), 378-399.ÌýÌý
• An improved Gregory-like method for 1-D quadrature (BF and J.A. Reeger), Numerische Mathematik, 141 (2019), 1-19.Ìý
• 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),ÌýJournal of Computational Physics, 374 (2018), 996-1016.ÌýÌýÌý
• Numerical quadrature over smooth surfaces with boundaries (J.A. Reeger and BF), Journal of Computational Physics, 355 (2018), 176-190.ÌýÌý
• Using radial basis function-generated finite differences (RBF-FD) to solve heat transfer equilibrium problems in domains with interfacesÌý(B. Martin and BF), Eng. Anal. w. Boundary Elements, 79 (2017), 38-48.Ìý
• (M. Fasondini, BF, J.A.C. Weideman),Ìý Journal of Computational Physics, 344 (2017), 36-50.Ìý
• 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.Ìý
• On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs (V. Bayona, N. Flyer, BF and G.A. Barnett),ÌýÌýJournal of Computational Physics, 332 (2017), 257-273.ÌýÌý
• Stable computations with flat radial basis functions using vector-valued rational approximations (G.B. Wright and BF),ÌýJournal of Computational Physics, 331 (2017), 137-156.ÌýÌý
• Fast calculation of Laurent expansions for matrix inverses, Journal of Computational Physics, 326 (2016), 722-732.
• On the role ofÌý polynomials in RBF-FD approximations: I. Interpolation and accuracy (N. Flyer, BF, G.A. Barnett and V. Bayona),ÌýJournal of Computational Physics, 321 (2016), 21-38.
• Numerical quadrature over the surface of a sphereÌý(J.A. Reeger and BF), Studies in Applied Math.Ìý137 (2015), 174-188.
• Solving PDEs with radial basis functionsÌý(BF and N. Flyer),ÌýActa NumericaÌý24 (2015), 215-258.
• A computational overview of the solution space of the imaginary Painlevé II equationÌý(BF and J.A.C. Weideman),ÌýPhysica D, 309 (2015), 108-118.
• Seismic modeling with radial basis function-generated finite differences (RBF-FD)Ìý(B. Martin, BF and A. St-Cyr),ÌýGeophysics, 80, No 4 (2015), T137-T146.
• Fast generation of 2-D node distributions for mesh-free PDE discretizationsÌý(BF and N. Flyer),ÌýComp. Math. Applic. 69 (2015), 531-544.Ìý(Files to run the main test problem:Ìýmain_script.m,Ìýnode_placing.m,Ìýradius_trui.m,Ìýtrui.png.)
• Stability ordinates of Adams predictor-corrector methodsÌý(M. Ghrist, BF and J.A. Reeger),ÌýBIT, 55 (2015), 733-750.
• Painlevé IV: A numerical study of the fundamental domain and beyondÌý(J.A. Reeger and BF),ÌýÌýPhysica DÌý280-281 (2014), 1-13.
• Radial basis function-generated finite differences: A mesh-free method for computational geosciencesÌý (N. Flyer, G.B. Wright and BF),ÌýHandbook of Geomathematics, 2014, Springer.
• A computational exploration of the second Painlevé equationÌý(BF and J.A.C. Weideman),ÌýFound. Comput. Math., 14 (2014), 985-1016.
• Some observations regarding steady laminar flows past bluff bodiesÌý(BF and A. Elcrat),ÌýPhil. Trans. Roy. Soc. London, Ser. A, 372 (2014), Article 20130353.
• On spherical harmonics based numerical quadrature over the surface of a sphereÌý(BF and J.M. Martel),ÌýAdv. Comp. Math., 40 (2014), 1169-1184.
• Inverting non-linear dimensionality reduction with scale-free radial basis interpolationÌý(N.D. Monnig, BF and F.G. Meyer),ÌýAppl. Comput. Harm. Anal., 37 (2014), 162-170.
• Ìý(C-I.R. Davis and BF),ÌýComplex Variables and Elliptic Equations, 59 (2014), 564-577.
• Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functionsÌý(E. Larsson, E. Lehto, A. Heryodono and BF), SIAM J. Sci.ÌýComp. 35 (2013), A2096-A2119.
• Stable calculation of Gaussian-based RBF-FD stencilsÌý(BF, E. LehtoÌýand C. Powell),ÌýComp. Math. Applic.,Ìý65 (2013), 627-637.
• Painlevé IV with both parameters zero: A numerical studyÌýÌý(J. Reeger and BF),ÌýStudies in Applied Math., 130 (2013), 108-133.
• Two results concerning the stability of staggered multistep methodsÌý(M. Ghrist and BF),ÌýSIAM J. Num. Anal.50 (2012), 1849-1860.
• A numerical implementation of Fokas boundary integral approach: Laplace's equation on a polygonal domainÌý(BF and N. Flyer),ÌýRoyal Society Proc. Series A.Ìý467 (2011), 2983-3003.
• A numerical methodology for the Painlevé equationsÌý(BF and J.A.C. Weideman),ÌýJournal of Computational Physics, 230 (2011), 5957-5973.
• Radial basis functions: Developments and applications to planetary scale flowsÌý(N. Flyer and BF),ÌýComputers and Fluids, 46 (2011), 23-32.
• Stabilization of RBF-generated finite difference methods for convective PDEsÌý(BF and E. Lehto),ÌýJournal of Computational Physics, 230 (2011), 2270-2285.
• Stable computations with Gaussian radial basis functionsÌý(BF, E. Larsson and N. Flyer),ÌýÌýSIAM J. Sci. Comput. 33 (2011), 869-892.Ìý
• Comparisons between different implementations of Newton's method for a nonlinear Poisson-type PDE (BF, N.ÌýFlyer and J.M. Russell), to be submitted.
• The Gibbs phenomenon for radial basis functionsÌý (BF and Natasha Flyer), inÌýThe 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 phenomenonÌý (T.A. Driscoll and BF), inÌýThe 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,ÌýJournal of Computational and Applied Mathematics,Ìý233 (2010), 2831-2840.
• Evolution of solitary waves in a two-pycnocline system (M. Nitsche, P.D. Weideman, R. Grimshaw, M. Ghrist and BF),ÌýÌýJ. Fluid Mech. 642 (2010), 235-277.
• Comparisons between pseudospectral and radial basis function derivative approximationsÌý(BF, N. Flyer and J.M. Russell),ÌýIMA Journal of Numerical Analysis, 30 (2010), 149-172.Ìý
• Magnetic relaxation in the solar coronaÌýÌý (K. Miller, BF, N. Flyer and B.C. Low),ÌýThe Astrophysical Journal,Ìý690 (2009), 720-733.
• Steady axisymmetric vortex flows with swirl and shearÌý(A. Elcrat, BF and K. Miller),ÌýJ. Fluid Mech, 613 (2008), 395-410.Ìý
• On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphereÌý(BF and C. Piret),ÌýJournal of Computational Physics, 227 (2008), 2758-2780.
• Locality properties of radial basis function expansion coefficients for equispacedÌýinterpolationÌýÌý (BF, Natasha Flyer, S. Hovde and C. Piret),ÌýIMA Journal of Numerical Analysis, 28 (2008), 121-142.
• A stable algorithm for flat radial basis functions on a sphereÌý(BF and C. Piret),ÌýSIAM J. Sci. Comp.Ìý30 (2007), 60-80.
• The Runge phenomenon and spatially variable shape parameters in RBF interpolationÌý(BF and J. Zuev),ÌýComp. Math. Applic.,Ìý54 (2007), 379-398.
• Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equationsÌý(BF, J. Zuev and J. Lee),ÌýJournal of Computational and Applied Mathematics, 200 (2007), 178-192.
• A pseudospectral fictitious point method for high order initial-boundary value problemsÌý,ÌýSIAM J. Sci. Comp.Ìý28 (2006), 1716-1729.
• A new class of oscillatory radial basis functionsÌý(BF, E. Larsson and G. Wright),ÌýComputers and Mathematics with ApplicationsÌý51 (2006), 1209-1222.
• ScatteredÌý node compact finite difference-type formulas generated from radial basis functionsÌý(G. Wright and BF),ÌýJournal of Computational PhysicsÌý212 (2006), 99-123.
• Stability of vortices in equilibrium with a cylinderÌýÌý (A. Elcrat, BF, and K. Miller),ÌýÌýJ. Fluid. Mech.Ìý544 (2005), 53-68.
• Magnetic-field confinement in the solar corona. II. Field-plasma interactionÌý(N. Flyer, BF, S. Thomas and B.C. Low),ÌýThe Astrophysical JournalÌý631 (2005), 1239-1259.
• Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite gridsÌý(BF and N. Flyer).ÌýAdvances in Computational MathematicsÌý23 (2005), 5-20.
• Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functionsÌý(E. Larsson and BF).ÌýComputers and Mathematics with ApplicationsÌý49 (2005), 103-130.
• Magnetic-field confinement in the solar corona. I. Force-free magnetic fieldsÌý(N. Flyer, BF, S. Thomas and B.C. Low).ÌýThe Astrophysical JournalÌý606 (2004), 1210-1222.
• Stable computation of multiquadric interpolants for all values of the shape parameterÌý(BF and G. Wright).ÌýComputers and Mathematics with ApplicationsÌý48 (2004), 853-867.
• Some unconditionally stable time stepping methods for the 3-D Maxwell's equationsÌý(J. Lee and BF).ÌýJournal of Computational and Applied MathematicsÌý166 (2004), 497-523.
• Some observations regarding interpolants in the limit of flat radial basis functionsÌý(BF, G. Wright andÌýE. LarssonÌý),ÌýComputers and Mathematics with ApplicationsÌý47 (2004), 37-55.
• On the nature of initial-boundary value solutions for dispersive equationsÌý(N. Flyer and BF).ÌýSIAMÌýJ. Appl. Math.64 (2003), 546-564.
• A numerical study of some radial basis function based solution methods for elliptic PDEsÌý(E. Larsson and BF),ÌýComputers and Mathematics with Applications, 46 (2003), 891-902.
• Accurate numerical resolution of transients in initial-boundary value problems for the heat equationÌý(N. Flyer and BF),ÌýJournal of Computational PhysicsÌý184/2 (2003), 526-539.
• Some numerical techniques for Maxwell's equations in different types of geometriesÌý,ÌýTopics 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 equationsÌý(J. Lee and BF),ÌýJournal of Computational and Applied MathematicsÌý158 (2003), 485-505.
• A short proof of the unconditional stability of the ADI-FDTD schemeÌý,ÌýCU APPM Preprint #472.
• Interpolation in the limit of increasingly flat radial basis functionsÌý(T.A. Driscoll and BF),ÌýComputers and Mathematics with Applications,Ìý43(2002), 413-422.
• Observations on the behavior of radial basis functions near boundariesÌý(BF, T.A. Driscoll, G. Wright and R. Charles),ÌýComputers and Mathematics with ApplicationsÌý43(2002), 473-490.
• Some steady axisymmetric vortex flows past a sphereÌý(A. Elcrat, BF and K. Miller),ÌýJ. Fluid Mech. 433 (2001), 315-328.
• A Padé-based algorithm for overcoming the Gibbs' phenomenonÌý(T.A. Driscoll and BF),ÌýNumerical AlgorithmsÌý26 (2001), 77-92.
• Note on nonsymmetric finite differences for Maxwell's equationsÌý(T.A. Driscoll and BF),ÌýJ. Comput. Phys.161 (2000), 723-727.
• Staggered time integrators for wave equationsÌý(BF, M. Ghrist and T.A. Driscoll),ÌýÌýSIAMÌýJ. Num. Anal. 38 (2000), 718-741.
• Some steady vortex flows past a circular cylinderÌý(A. Elcrat, BF, M. Horn and K. Miller),ÌýÌýJ. Fluid. Mech.Ìý409 (2000), 13-27.
• A fast spectral algorithm for nonlinear wave equations with linear dispersionÌý(BF and T.A. Driscoll),ÌýJ. Comput. Phys.Ìý155 (1999), 456-467.
• Spatial finite difference approximations for wave-type equationsÌý(BF and M. Ghrist).ÌýSIAMÌýJ. Num. Anal. 37 (1999), 105-130.
• Block pseudospectral methods for Maxwell's equations: II. Two-dimensional, discontinuous-coefficient caseÌý(T.A. Driscoll and BF).ÌýSIAMÌýJ. Sci. Comput.Ìý21 (1999), 1146-1167.
• On the chance of freak waves at seaÌý(B.S. White and BF).ÌýJ. Fluid. Mech.Ìý355 (1998), 113-138.
• A block pseudospectral method for Maxwell's equations: I. One-dimensional caseÌý(T.A. Driscoll and BF).ÌýJ. Comput. Phys.Ìý140 (1998), 47-65.
• Calculation of weights in finite difference formulas.ÌýSIAMÌýReviewÌý40 Nr 3, (1998), 685-691.
• Comparison of finite difference- and pseudospectral methods for convective flow over a sphereÌý(BF andÌýD. Merrill).Geophys. Res. Lett.Ìý24, No 24 (1997), 3245-3248. (Summarizes Dave Merrill'sÌýMaster's Thesis).
• A compact fourth order finite difference scheme for the steady incompressible Navier-Stokes equationsÌý, (M. Li, T. Tang and BF),ÌýInt. J. for Numerical Methods in Fluids, 20 (1995), 1137-1151.
• Computing steady incompressible flows past blunt bodies - A historical overviewÌý, inÌýNumerical Methods for Fluid Dynamics IVÌý(Ed. M.J. Baines and K.W. Morton) Oxford Univ. Press (1993), 115-134.
• Steady incompressible flow past a row of circular cylindersÌý,ÌýJ. Fluid Mech. 225 (1991), 655-671.
• Generation of finite difference formulas on arbitrarily spaced gridsÌý,ÌýMathematics of Computation, 51 (1988), 699-706.
• Steady viscous flow past a sphere at high Reynolds numbersÌý,ÌýJ. 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,ÌýJournal of Computational Physics,Ìý61 (1985), 297-320.
• Algorithm 579, CPSC: Complex Power Series Coefficients,ÌýACM Transactions in Mathematical Software,Ìý7 (1981), 542-547.
• Numerical differentiation of analytic functions,ÌýACM Transactions in Mathematical Software,Ìý7 (1981), 512-526.
• A numerical study of steady viscousÌýflow past a circular cylinderÌý,ÌýJ. Fluid Mech. 98 (1980), 819-855.
• A numerical method for conformal mappingsÌý,ÌýSIAM J. Sci. Stat. Comp.,Ìý1(1980), 386-400.
• A numerical and theoretical study of certain nonlinear wave phenomenaÌý(BFÌýand G.B. Whitham),ÌýPhil. Trans. Royal Soc. London, Ser A, Vol 289 (1978), 373-404.
• A numerical study of 2-D turbulenceÌý,ÌýJournal of Computational Physics,Ìý25 (1977), 1-31.
• On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equationÌý,ÌýMathematics of Computation, 27 (1973), 45-57.
• A method for acceleration of the convergence of infinite seriesÌý, (A. Beckman, BF and A. Tengvald), BIT, 9 (1969), 78-80.