Mathematics
- Mathematical Software and Scientific Computing
- High Order Numerical Methods for Partial Differential
Equations: Pseudospectral and Radial Basis Function methods
- Approximation Theory: Spectral Postprocessing, Orthogonal Polynomials,
Radial Basis Functions, Wavelets, etc.
Software
- Radial Basis Function Toolbox
- version 1.0 (May 27, 2016, Matlab)
- version 1.1 (April 20, 2017, Matlab)
- version 2.0 (June 2022, Python)
- Matlab Postprocessing Suite -
removal of Gibbs oscillations from the Chebyshev and Fourier
approximation of discontinuous functions.
- version 1.0 (2003, Java)
- version 2.0 (2009, Matlab)
- Spectral Methods with
Postprocessing for Numerical Hyperbolic Heat Transfer. Numerical Heat Transfer, Part A, vol. 43 no. 7 (2003), p.
717-730. DOI: 10.1080/713838126
- Chebyshev Super Spectral Viscosity
method for a Fluidized Bed Model. Journal of Computational Physics, 186/2, p. 630-651 (2003). DOI: 10.1016/S0021-9991(03)00089-5
- Chebyshev Super Spectral Viscosity
method for a Two-Dimensional Fluidized Bed Model International Journal of Numerical Methods in Fluids, vol. 42,
p. 249-263. DOI: 10.1002/fld.485
- The Spectral Signal Processing Suite. ACM Transactions on Mathematical
Software, vol. 29, no. 2, June 2003, p. 1-23. DOI: 10.1145/779359.779364
- The Method of Characteristics with applications to Conservation Laws.
Journal of Online Mathematics and Applications (JOMA), Volume 3, February
2003. [applet], [pdf version]
- Adaptive Radial Basis Function Methods for time dependent Partial Differential Equations. Applied Numerical Mathematics, vol.
54, issue 1, p. 79-94, June 2005. DOI: 10.1016/j.apnum.2004.07.004
- Integrated Multiquadric
Radial Basis Function Approximation Methods. Computers and Mathematics with Applications, vol. 51, no. 8,
April 2006, p. 1283-1296. DOI: 10.1016/j.camwa.2006.04.014
- Digital Total Variation
Filtering as Postprocessing for Pseudospectral Methods for Conservation Laws. Numerical Algorithms,
vol. 41, p. 17-33, 2006. DOI: 10.1007/s11075-005-9003-5
- Digital Total Variation Filtering as postprocessing for Radial Basis Function Approximation Methods. Computers and Mathematics with Applications, vol. 52, p.
1119-1130, Sept. 2006. DOI: 10.1016/j.camwa.2006.02.013
- Chebyshev Interpolation: an
interactive tour Journal of Online Mathematics and Applications, September 2006. [html],
[pdf], [applets: Chebyshev approximation, Chebyshev
polynomial, Runge phenomenon, exponential filter]
- A Numerical Study of the Accuracy and Stability of Symmetric and Asymmetric RBF Collocation methods for Hyperbolic PDEs. Numerical Methods for Partial Differential Equations, vol. 24, no. 2, p. 670 -
686, 2008. DOI: 10.1002/num.20290
- A Pseudospectral Method with Edge Detection Free Postprocessing for 2d Hyperbolic Heat Transfer. Numerical Heat Transfer, Part B:
Fundamentals vol. 54, no. 1, 2008, p. 52-61. DOI: 10.1080/10407790802122485
- Edge Detection Free Postprocessing for Pseudospectral Approximations. Journal of Scientific Computing, vol. 51, no. 1, p. 49-61,
2009. DOI: 10.1007/s10915-009-9287-z
- A Random Variable Shape Parameter Strategy for Radial Basis Function Approximation Methods (with D. Sturgill). Engineering
Analysis with Boundary Elements, vol. 33, p. 1239-1245,
2009. DOI: 10.1016/j.enganabound.2009.07.003
- Multiquadric Radial Basis Function Approximation Methods for the Numerical
Solution of Partial Differential Equations (with E. J. Kansa). Advances in Computational Mechanics, volume 2, 2009. ISSN: 1940-5820. (Matlab
m-files, pdf).
- A Numerical Study of Generalized Multiquadric Radial Basis Function Interpolation (with M. Chenoweth). SIAM Undergraduate Research Online Volume
2, Issue 2, 2009. DOI: 10.1137/09S01040X
- Algorithm 899: The
Matlab Postprocessing Toolkit. ACM Transactions on Mathematical Software, vol. 37, no. 1, 2010. DOI: 10.1145/1644001.1644011
- Radial
Basis Function approximation methods with extended precision floating point arithmetic. Engineering Analysis with Boundary Elements, Volume 35, Issue 1, January
2011, Pages 68-76. DOI: 10.1016/j.enganabound.2010.05.011
- Stability propertiesof a predictor-corrector implementation of an implicit linear multistep method (with C. Meador). Involve, vol. 4,
no. 1, p. 43-51, 2011. DOI: 10.2140/involve.2011.4.43
- On the numerical solution of chaotic dynamical systems using extended precision floating point arithmetic and very high order numerical methods (with C. Meador). Nonlinear
Analysis: Modelling and Control, vol. 16, no. 3, 2011, p. 340-352.
- A Linear System-Free Gaussian RBF Method for the Gross-Pitaevskii equation on unbounded domains. Numerical Methods
for Partial Differential Equations, volume 28, number 2, p. 389-401, 2012. DOI: 10.1002/num.20623
- A Local Radial Basis Function method for Advection-Diffusion-Reaction equations on complexly shaped domains. Applied Mathematics
and Computation, volume 218, p. 9853-9865, 2012. Matlab files. DOI: 10.1016/j.amc.2012.03.062
- Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation.
Engineering Analysis with Boundary Elements, volume 44, p. 76-86, July 2014. (source code)
DOI: 10.1016/j.enganabound.2014.04.019
- Filtered Leapfrog Time Integration with Enhanced Stability Properties (with A. Aluthge, and R.
Estep). Journal of Applied Mathematics and Physics, 4, p. 1354-1370, 2016. DOI: 10.4236/jamp.2016.47145. (source code)
- An examination of evaluation algorithms for the RBF methods
(with S. Cogar). Engineering Analysis with Boundary Elements, volume 75, p. 36-45, February 2017. DOI:
10.1016/j.enganabound.2016.11.006, (source code)
- The Matlab Radial Basis Function Toolkit. Journal of Open Research Software, volume 5, issue 1, page 8,
2017. (software web page) DOI: 10.5334/jors.131
- Radial Basis Function Methods - the case of symmetric domains.
Numerical Methods for Partial Differential Equations, 2018. DOI:10.1002/num.22272
- A rational radial basis function method for accurately resolving discontinuities and steep gradients (with Y.
Bai). Applied Numerical Mathematics 130, p. 131-142, 2018. DOI: 10.1016/j.apnum.2018.04.001
- Local Radial Basis Functios methods: comparison, improvements, and implementation (2022)
Bibtex entries for publications above, ORCID ID:
0000-0003-0989-412X, Researchgate, doi2bib
Technical Reports
Books
- (Textbook) Numerical Analysis and Scientific Computing with
Python (anticipated completion date: June 2024).
- (Research Monograph) Radial Basis Function Methods for the numerical
solution of Partial Differential Equations (completetion date
TBD).
- Spring 2016: Partial Differential Equations (MTH 415/615), Calculus 2
- Fall 2016: Numerical Analysis 1 - Introduction (MTH 443/643), Numerical Analysis 3 - PDEs (MTH 667)
- Spring 2017: no classes, Sabbatical leave
- Fall 2017: Numerical Analysis 1 - Introduction (MTH 443/643),
Calculus 2
- Spring 2018: Numerical Analysis 2 - Linear Algebra (MTH
442/642), Partial Differential Equations (MTH 415/615)
- Fall 2018: Numerical Analysis 1 - Introduction (MTH 443/643),
Numerical Analysis 3 - PDEs (MTH 667)
- Spring 2019: Numerical Analysis 2 - Linear Algebra (MTH
442/642), Complex Analysis (MTH 640), Ordinary Differential
Equations (MTH 335)
- Fall 2019: Numerical Analysis 1 - Introduction (MTH 443/643),
Calculus 2
- Spring 2020: Numerical Analysis 3 - Numerical PDEs, Calculus 2
- Fall 2020: Numerical Analysis 1 - Introduction, Calculus 2
- Spring 2021: Numerical Analysis 2, Calculus 2
- Fall 2021: Numerical Analysis 1, Calculus 2
- Spring 2022: Numerical 2 - Linear Algebra (MTH 642), Calculus 2 (MTH 230), Discrete Structures (MTH 220)
- R.-R.Mace - Rational Reconstruction of
Discontinuous Functions to remove Gibb's oscillations
- Mike Smith - Radial Basis Function methods in Numerical Relativity, 2009
- Derek Sturgill - Variable shape parameter strategies in RBF methods
- Clyde Meador- Numerical methods for Chaotic Dynamical Systems
- Maggie Chenoweth - Local Radial Basis Function methods and pattern formation in Mathematical Biology
- Jack Goodman - Numerical methods for dense, ill-conditioned, symmetric positive definite linear systems, 2014
- Yikun Bai - Rational RBF methods, 2016
Undergraduate
- Caitlyn Davis-McDaniel - A Matlab graphical user interface to
illustrate iterative methods for solving nonlinear equations
- Amy Sietz - RBF methods and Bose-Einstein Condensates
- Clyde Meador - Numerical methods for Chaotic Dynamical Systems
- Maggie Chenoweth - generalized Multiquadric Radial Basis Function interpolation
- Brittany Whited - Mathematical Biology: Modelling Gravitropic Curvature in Pea
Stems
- senior capstone project, 2012
- Roger Estep - Filtered leapfrog time differencing
- Andrew Henrichs - Numerical methods for a chaotic dynamical system from chemistry
- senior capstone project 2014
- Shawn Cheeks
- Sam Cogar
- Daniel Davis
- Dillon Buskirk
- Emily Sutherland
- senior capstone 2019 - Radial Basis Function viscosity methods for hyperbolic PDEs
- Christian Gunnoe
- senior capstone 2020 - The Lambert W function
- Ian Waddell
- senior capstone 2021 - Numerical solution of the Blasius flow problem
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