Papers by Graeme Fairweather
High-order orthogonal spline collocation method with graded meshes for two-dimensional fractional evolution integro-differential equation
International Journal of Computer Mathematics, 2021
ACM Transactions on Mathematical Software, Dec 1, 1978
A summary is made of a recent investigation of Romberg quadrature routines In that study the perf... more A summary is made of a recent investigation of Romberg quadrature routines In that study the performance of several existing Romberg routines on a set of test problems was examined and three new routines, constructed to exhibit the effects of sequence choice and cautious extrapolation, were described Conclusions are reached concerning the form of an acceptable nonadaptive Romberg routine.
Numerical Methods for Partial Differential Equations, Dec 28, 2014
We formulate and analyze a novel numerical method for solving a time-fractional Fokker-Planck equ... more We formulate and analyze a novel numerical method for solving a time-fractional Fokker-Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time-stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence.
ACM Transactions on Mathematical Software, Jun 1, 1992
techniques based on piecewise polynomial (that is, spline) collation at Gaussian .
Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type
Numerische Mathematik, May 1, 1967
... The (ID) method is sixth order correct for solving LAI'LACE'S equation and converge... more ... The (ID) method is sixth order correct for solving LAI'LACE'S equation and converges for p~ 3-In this case, a parameter sequence can be obtained and is described in the next section. It can be seen from the values of K given ...
A New Alternating Direction Method for Parabolic Equations in Three Space Variables
Journal of the Society for Industrial and Applied Mathematics, Dec 1, 1965
... _r8y2 + 1)um*+* = -ray Um + UM+1 (-r82 + l)urn, = -r + Ur + UM+ where ax, 8by, &, are the... more ... _r8y2 + 1)um*+* = -ray Um + UM+1 (-r82 + l)urn, = -r + Ur + UM+ where ax, 8by, &, are the usual central difference operators in the x, y, and z directions respectively, um, um+1 are the values of u at the nodes (iAx, jAy, kAz) at times mAt and (n + 1 )At respectively ... (7) (1 + Aay 2)u ...
Mathematics of Computation, Sep 1, 1984
. Replace Eq. (A18) with , / 2 X IX2 X3 31A4 This correction has no effect on any of our results,... more . Replace Eq. (A18) with , / 2 X IX2 X3 31A4 This correction has no effect on any of our results, since the code used the correct equation.
Journal of Computational and Applied Mathematics, Jun 1, 1993
Bialecki, B. and G. Fairweather, Matrix decomposition algorithms for separable elliptic boundary ... more Bialecki, B. and G. Fairweather, Matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions, Journal of Computational and Applied Mathematics 46 ( 1993) 369-386. In recent years, several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite-difference, finite-element Galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle. The success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two-point boundary value problems in one space dimension. The primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework. Particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation, that is, spline collocation at Gauss points. All of the methods discussed in this paper are modular and possess a great deal of natural parallelism.
Journal of the Acoustical Society of America, Dec 1, 1998
The method of fundamental solutions ͑MFS͒ is applied to acoustic scattering and radiation for axi... more The method of fundamental solutions ͑MFS͒ is applied to acoustic scattering and radiation for axisymmetric bodies and boundary conditions. The fundamental solution of the governing equation and its normal derivative, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals, which are evaluated using library software. The method is tested on several problems from the literature and the results compared with existing solutions. Numerical experiments demonstrate that the fictitious eigenfrequency problem which is encountered with the boundary element method is not present in the MFS.
International Journal for Numerical Methods in Engineering, Jul 1, 1988
A new fundamental solutions method for the numerical solution of two-dimensional biharmonic probl... more A new fundamental solutions method for the numerical solution of two-dimensional biharmonic problems is described. In this method, which is based on the Almansi representation of a biharmonic function in the plane, the approximate solution is expressed in terms of fundamental solutions of Laplace's equation, and is determined by a least squares fit of the boundary conditions. The results of numerical experiments which demonstrate the efficacy of the method are presented. N
SIAM Journal on Numerical Analysis, Apr 1, 1986
This paper surveys the different design issues, from mathematical model to silicon, involved on t... more This paper surveys the different design issues, from mathematical model to silicon, involved on the design of integrated circuits for the generation of chaotic behavior.
Algorithms for Almost Block Diagonal Linear Systems
Siam Review, 2004
Codes for solving systems of ordinary differential equations for use in the method of lines for p... more Codes for solving systems of ordinary differential equations for use in the method of lines for partial differential equations (PDEs) usually provide only a banded system solver. In this context, a frequently occurring structure is almost block diagonal (ABD). Solving ABD systems by imposing banded structure introduces fill-in and is inefficient. Though robust, efficient ABD software has been developed and used in packages for solving boundary value problems with separated boundary conditions for ordinary differential equations (BVODEs), it has not been generally exploited in PDE software. The situation with bordered almost block diagonal system software for BVODEs with nonseparated boundary conditions is less satisfactory. This survey draws on material from a variety of sources, particularly [P. Amodio et al., Numer. Linear Algebra Appl., {7} (2000), pp. 275--317] and [B. Garrett and I. Gladwell, J. Comput. Methods Sci. Engrg., {1} (2001), pp. 75--98] and the references therein.
Spline Collocation Methods for a Class of Hyperbolic Partial Integro-Differential Equations
SIAM Journal on Numerical Analysis, Apr 1, 1994
Spline collocation methods are proposed for the spatial discretization of a class of hyperbolic p... more Spline collocation methods are proposed for the spatial discretization of a class of hyperbolic partial integro-differential equations arising in the theory of linear viscoelasticity. For problems in two space variables, error estimates are derived for the continuous-time orthogonal spline collocation method and three discrete-time orthogonal spline collocation methods. The continuous-time modified cubic spline collocation method for problems in one space variable is also analyzed.
Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions
Journal of the Acoustical Society of America, Apr 1, 1992
The method of fundamental solutions (MFS) is a boundary method for the numerical solution of cert... more The method of fundamental solutions (MFS) is a boundary method for the numerical solution of certain elliptic boundary value problems. In the MFS, the approximate solution is a linear combination of fundamental solutions of the governing partial differential equation, with singularities placed outside the domain of the problem. In the present paper, the MFS is applied to acoustic scattering in fluids. The singularities are allowed to move during the solution process from arbitrary locations to more optimal locations. Numerical results demonstrate that the ‘‘fictitious eigenfrequency’’ difficulty encountered with the boundary element method (BEM) is not present in the MFS. In addition, MFS results obtained by the use of fixed singularities are presented for scattering of waves in elastic solids.
ACM Transactions on Mathematical Software, Sep 1, 1983
Orthogonal Spline Collocation Methods for Some Partial Integrodifferential Equations
SIAM Journal on Numerical Analysis, Jun 1, 1992
... Under these assumptions, it is easy to see that, for sufficiently large so, lim a(i)(t)e-80t ... more ... Under these assumptions, it is easy to see that, for sufficiently large so, lim a(i)(t)e-80t = 0, j = 0, 1. t -OO Page 9. ... Page 10. 764 YI YAN AND GRAEME FAIRWEATHER ... Now, applying the triangle inequality and Lemma 4.3, we have llw - Qwll < |(I - Ph)wII + II(Q - ph)wII ...
Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for solving parabolic and elliptic equations
Numerische Mathematik, Dec 1, 1964
Atmospheric Environment, 1995
Limitations on comprehensive tropospheric chemistry/transport models are discussed within the con... more Limitations on comprehensive tropospheric chemistry/transport models are discussed within the context of a ~et of issues currently facing the environmental scientific and policy-making communities. A number of central improvements are discussed in a prioritized manner, with consideration of the key progress nece ~sary to include feedback processes between meteorology and chemistry, aerosol formatiot~ in cloud development with subsequent effects on wet removal, dry deposition and surface exchangeprocesses, and impacts of chemical perturbations on radiation, climate, and weather. These improvements would result in a "third-generation model". The computational framework for this code is outlined, and estimates of required computer resources presented.
Nonlinear Analysis-theory Methods & Applications, Aug 1, 1988
Nonlinear parabolic and hyperbolic partial integro-differential equations, discrete time Galerkin... more Nonlinear parabolic and hyperbolic partial integro-differential equations, discrete time Galerkin methods, discrete time collocation methods, Crank-Nicolson Galerkin method, Crank-Nicolson collocation method, optimal error estimates.
Engineering Analysis With Boundary Elements, Jul 1, 2003
The development of the method of fundamental solutions (MFS) and related methods for the numerica... more The development of the method of fundamental solutions (MFS) and related methods for the numerical solution of scattering and radiation problems in fluids and solids is described and reviewed. A brief review of the developments and applications in all areas of the MFS over the last five years is also given. Future possible areas of applications in fields related to scattering and radiation problems are identified.
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Papers by Graeme Fairweather