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767
ImplicitExplicit Methods For TimeDependent PDEs
 SIAM J. NUMER. ANAL
, 1997
"... Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection ..."
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Cited by 177 (6 self)
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Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reactiondiffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods. For the prototype linear advectiondiffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay of high freque...
Barycentric Lagrange Interpolation
, 2004
"... Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation. ..."
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Cited by 133 (6 self)
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Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.
FFTs for the 2SphereImprovements and Variations
 JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
, 2003
"... Earlier work by Driscoll and Healy [18] has produced an efficient algorithm for computing the Fourier transform of bandlimited functions on the 2sphere. In this article we present a reformulation and variation of the original algorithm which results in a greatly improved inverse transform, and co ..."
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Cited by 127 (3 self)
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Earlier work by Driscoll and Healy [18] has produced an efficient algorithm for computing the Fourier transform of bandlimited functions on the 2sphere. In this article we present a reformulation and variation of the original algorithm which results in a greatly improved inverse transform, and consequent improved convolution algorithm for such functions. All require at most O(N log2 N)operations where N is the number of sample points. We also address implementation considerations and give heuristics for allowing reliable and computationally efficient floating point implementations of slightly modified algorithms. These claims are supported by extensive numerical experiments from our implementation in C on DEC, HP, SGI and Linux Pentium platforms. These results indicate that variations of the algorithm are both reliable and efficient for a large range of useful problem sizes. Performance appears to be architecturedependent. The article concludes with a brief discussion of a few potential applications.
Comparing Solution Methods for Dynamic Equilibrium Economies
 Journal of Economic Dynamics and Control
, 2006
"... This paper compares solution methods for dynamic equilibrium economies. We compute and simulate the stochastic neoclassical growth model with leisure choice using Undetermined Coefficients in levels and in logs, Finite Elements, Chebyshev Polynomials, Second and Fifth Order Perturbations and Value F ..."
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Cited by 103 (32 self)
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This paper compares solution methods for dynamic equilibrium economies. We compute and simulate the stochastic neoclassical growth model with leisure choice using Undetermined Coefficients in levels and in logs, Finite Elements, Chebyshev Polynomials, Second and Fifth Order Perturbations and Value Function Iteration for several calibrations. We document the performance of the methods in terms of computing time, implementation complexity and accuracy and we present some conclusions about our preferred approaches based on the reported evidence.
Fourthorder time stepping for stiff PDEs
 SIAM J. Sci. Comput
, 2005
"... Abstract. A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison ..."
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Cited by 95 (3 self)
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Abstract. A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential timedifferencing (ETD) scheme against the competing methods of implicitexplicit differencing, integrating factors, timesplitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
A MATLAB differentiation matrix suite
 ACM TOMS
, 2000
"... A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolan ..."
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Cited by 68 (3 self)
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A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.
An efficient spectralprojection method for the Navier–Stokes equations in cylindrical geometries I. Axisymmetric cases
 J. Comput. Phys
, 1998
"... An efficient and accurate numerical scheme is presented for the threedimensional Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a spectralGalerkin approximation for the space variables and a secondorder projection scheme for time. The new spectralprojection ..."
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Cited by 58 (20 self)
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An efficient and accurate numerical scheme is presented for the threedimensional Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a spectralGalerkin approximation for the space variables and a secondorder projection scheme for time. The new spectralprojection scheme is implemented to simulate unsteady incompressible flows in a cylinder. c ○ 2002 Elsevier Science (USA) Key Words: spectralGalerkin method; projection method; Navier–Stokes; rotating waves.
A novel pricing method for European options based on Fouriercosine series expansions
 SIAM J. SCI. COMPUT
, 2008
"... Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fouriercosine expansion of the density function. In most cases, ..."
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Cited by 55 (14 self)
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Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fouriercosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including Lévy processes and the Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a followup paper we will present its application to options with earlyexercise features.
Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs
, 1997
"... . Let P = fP 0 ; : : : ; Pn\Gamma1 g denote a set of polynomials with complex coefficients. Let Z = fz 0 ; : : : ; z n\Gamma1 g ae C denote any set of sample points. For any f = (f 0 ; : : : ; fn\Gamma1 ) 2 C n the discrete polynomial transform of f (with respect to P and Z) is defined as the col ..."
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Cited by 49 (8 self)
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. Let P = fP 0 ; : : : ; Pn\Gamma1 g denote a set of polynomials with complex coefficients. Let Z = fz 0 ; : : : ; z n\Gamma1 g ae C denote any set of sample points. For any f = (f 0 ; : : : ; fn\Gamma1 ) 2 C n the discrete polynomial transform of f (with respect to P and Z) is defined as the collection of sums, f b f(P 0 ); : : : ; b f(Pn\Gamma1 )g, where f(P j ) = hf; P j i = P n\Gamma1 i=0 f i P j (z i )w(i) for some associated weight function w. These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of N orthogonal polynomials of degree at most N \Gamma 1 we give an O(N log 2 N) algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the N 2 operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a thre...
A stable algorithm for flat radial basis functions
 SIAM J. Sci. Comp
"... Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Rungetype oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstabl ..."
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Cited by 47 (6 self)
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Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Rungetype oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in case of nearflat basis functions, there will typically also be a separate point at which disasterous illconditioning enters. We introduce here a new method, RBFQR, which entirely eliminates such illconditioning, and we apply it in the special case when the data points are distributed over the surface of a sphere. This algorithm works even for thousands of node points, and it allows the RBF shape parameter to be optimized without the limitations imposed by stability concerns. Since interpolation in the flat RBF limit on a sphere is found to coincide with spherical harmonics interpolation, new insights are gained as to why the RBF approach (with nonflat basis functions) often is the more accurate of the two methods. Key words. Radial basis functions, RBF, shape parameter, sphere, spherical harmonics. 1. Introduction. Numerical