Fourier Spectral Methods for Degasperis–Procesi Equation with Discontinuous Solutions

In this paper, we develop, analyze and test the Fourier spectral methods for solving the Degasperis–Procesi (DP) equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The \(L^2\) stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method. By applying the Gegenbauer reconstruction technique as a post-processing method to the Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully. The numerical simulation results for different types of solutions of the nonlinear DP equation are provided to illustrate the accuracy and capability of the methods.     

Hermite–Sobolev orthogonal functions and spectral methods for second- and fourth-order problems on unbounded domains

Hermite spectral methods using Sobolev orthogonal/biorthogonal basis functions for solving second and fourth-order differential equations on unbounded domains are proposed. Some Hermite–Sobolev orthogonal/biorthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. The convergence is analyzed and some numerical results are presented to illustrate the effectiveness and the spectral accuracy of this approach.     

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

Reducing the Effects of Noise in Image Reconstruction

Fourier spectral methods have proven to be powerful tools that are frequently employed in image reconstruction. However, since images can be typically viewed as piecewise smooth functions, the Gibbs phenomenon often hinders accurate reconstruction. Recently, numerical edge detection and reconstruction methods have been developed that effectively reduce the Gibbs oscillations while maintaining high resolution accuracy at the edges. While the Gibbs phenomenon is a standard obstacle for the recovery of all piecewise smooth functions, in many image reconstruction problems there is the additional impediment of random noise existing within the spectral data. This paper addresses the issue of noise in image reconstruction and its effects on the ability to locate the edges and recover the image. The resulting numerical method not only recovers piecewise smooth functions with very high accuracy, but it is also robust in the presence of noise.     

Wave envelope technique for multimode wave guide problems

In this paper we propose a fast method for solving wave guide problems. In particular, we consider the guide to be inhomogeneous, and allow propagation of waves of higher-order modes. Such techniques have been handled successfully for acoustic wave propagation problems with single mode and finite length. This paper extends this concept to electromagnetic wave guides with several modes and infinite in length. The method is shown and results of computations are presented.Research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18107 while the first author was in residence at the ICASE, NASA Langley Research Center, Hampton, VA 23665-5225, and by NASA Grant No. NAG-1-624.     

Existence and numerical simulation of solutions for nonlinear fractional pantograph equations

In this paper a method based on Sinc approximation is developed for the numerical solution of a nonlinear fractional pantograph equation. In order to use Sinc approximation, the problem needs to have an analytic solution. So we investigated the existence and uniqueness of analytic solutions in the proposed domain. Single and double exponential transformations are used to approximate the solution. Some test problems are given to demonstrate the efficiency and applicability of the methods. The results are compared with some other existing numerical methods to show the performance and good accuracy of the methods. The numerical orders of convergence show that the method has exponential rate of convergence.     

Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Numerical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.     

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted \(L_2\) space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.     

Detection of Edges in Spectral Data III—Refinement of the Concentration Method

Edge detection from Fourier spectral data is important in many applications including image processing and the post-processing of solutions to numerical partial differential equations. The concentration method, introduced by Gelb and Tadmor in 1999, locates jump discontinuities in piecewise smooth functions from their Fourier spectral data. However, as is true for all global techniques, the method yields strong oscillations near the jump discontinuities, which makes it difficult to distinguish true discontinuities from artificial oscillations. This paper introduces refinements to the concentration method to reduce the oscillations. These refinements also improve the results in noisy environments. One technique adds filtering to the concentration method. Another uses convolution to determine the strongest correlations between the waveform produced by the concentration method and the one produced by the jump function approximation of an indicator function. A zero crossing based concentration factor, which creates a more localized formulation of the jump function approximation, is also introduced. Finally, the effects of zero-mean white Gaussian noise on the refined concentration method are analyzed. The investigation confirms that by applying the refined techniques, the variance of the concentration method is significantly reduced in the presence of noise. This work was partially supported by NSF grants CNS 0324957, DMS 0510813, DMS 0652833, and NIH grant EB 025533-01 (AG).     

An hp a priori error analysis of¶the DG time-stepping method for initial value problems

The Discontinuous Galerkin (DG) time-stepping method for the numerical solution of initial value ODEs is analyzed in the context of the hp-version of the Galerkin method. New a priori error bounds explicit in the time steps and in the approximation orders are derived and it is proved that the DG method gives spectral and exponential accuracy for problems with smooth and analytic time dependence, respectively. It is further shown that temporal singularities can be resolved at exponential rates of convergence if geometrically refined time steps are employed. Received: November 1999 / Accepted: January 2000     

Local tests for consistency of support hyperplane data

Support functions and samples of convex bodies in R ⁿ are studied with regard to conditions for their validity or consistency. Necessary and sufficient conditions for a function to be a support function are reviewed in a general setting. An apparently little known classical such result for the planar case due to Rademacher and based on a determinantal inequality is presented and a generalization to arbitrary dimensions is developed. These conditions are global in the sense that they involve values of the support function at widely separated points. The corresponding discrete problem of determining the validity of a set of samples of a support function is treated. Conditions similar to the continuous inequality results are given for the consistency of a set of discrete support observations. These conditions are in terms of a series of local inequality tests involving only neighboring support samples. Our results serve to generalize existing planar conditions to arbitrary dimensions by providing a generalization of the notion of nearest neighbor for plane vectors which utilizes a simple positive cone condition on the respective support sample normals.This work partially supported by the Center for Intelligent Control Systems under the U.S. Army Research Office Grant DAAL03-92-G-0115, the Office of Naval Research under Grant N00014-91-J-1004, and the National Science Foundation under Grant MIP-9015281.Partially supported by the National Science Foundation under grant IRI-9209577 and by the U.S. Army Research Office under grant DAAL03-92-G-0320     

Blocking for external graph searching

In this paper we consider the problem of using disk blocks efficiently in searching graphs that are too large to fit in internal memory. Our model allows a vertex to be represented any number of times on the disk in order to take advantage of redundancy. We give matching upper and lower bounds for completed-ary trees andd-dimensional grid graphs, as well as for classes of general graphs that intuitively speaking have a close to uniform number of neighbors around each vertex. We also show that, for the special case of grid graphs blocked with isothetic hypercubes, there is a provably better speed-up if even a small amount of redundancy is permitted.Support was provided in part by an IBM Graduate Fellowship, by NSF Research Grants CCR-9007851 and IRI-9116451, and by Army Research Office Grant DAAL03-91-G-0035.Support was provided in part by NSF Grants CCR-9003299, CCR-9300079, and IRI-9116843, and by NSF/DARPA Grant CCR-8908092.Support was provided in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF Research Grant CCR-9007851, and by Army Research Office Grant DAAL03-91-G-0035.     

A general analytical method for axisymmetric incompressible potential flow about bodies of revolution

A general analytical method is presented for calculating the flow field about bodies of revolution in incompressible potential flow as a sequence of elementary analytic functions (Fourier, Chebyshev and Legendre). For the cases of greatest interest to practical engineering (cusp and blunt trailing edges) comparison with an existing highly accurate numerical method shows that convergence is good. Only ten terms are required to give adequate accuracy for negligible computer usage on bodies with much more character than usually encountered. The theory can be used for design (inverse) operation to produce body shapes associated in a least squares sense with a desired input velocity. It is shown that, when used this way, again good results are obtained with remarkably few terms.     

An efficient parallel algorithm for shortest paths in planar layered digraphs

Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by giving efficient parallel algorithms for shortest paths in planar layered digraphs.We show that these graphs admit special kinds of separators calledone- way separators which allow the paths in the graph to cross it only once. We use these separators to give divide- and -conquer solutions to the problem of finding the shortest paths between any two vertices. We first give a simple algorithm that works in the CREW model and computes the shortest path between any two vertices in ann-node planar layered digraph in timeO(log² n) usingn/logn processors. We then use results of Aggarwal and Park [1] and Atallah [4] to improve the time bound toO(log² n) in the CREW model andO(logn log logn) in the CREW model. The processor bounds still remain asn/logn for the CREW model andn/log logn for the CRCW model.Support for the first and third authors was provided in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052, ARPA, Order 8225. Support for the second author was provided in part by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052 and ARPA Order 8225.     

A Semi-Lagrangian Method for Turbulence Simulations Using Mixed Spectral Discretizations

We present a semi-Lagrangian method for integrating the three-dimensional incompressible Navier–Stokes equations. We develop stable schemes of second-order accuracy in time and spectral accuracy in space. Specifically, we employ a spectral element (Jacobi) expansion in one direction and Fourier collocation in the other two directions. We demonstrate exponential convergence for this method, and investigate the non-monotonic behavior of the temporal error for an exact three-dimensional solution. We also present direct numerical simulations of a turbulent channel-flow, and demonstrate the stability of this approach even for marginal resolution unlike its Eulerian counterpart.     

Spectral element spatial discretization error in solving highly anisotropic heat conduction equation

This paper describes a study of the effects of the overall spatial resolution, polynomial degree and computational grid directionality on the accuracy of numerical solutions of a highly anisotropic thermal diffusion equation using the spectral element spatial discretization method. The high-order spectral element macroscopic modeling code SEL/HiFi has been used to explore the parameter space. It is shown that for a given number of spatial degrees of freedom, increasing polynomial degree while reducing the number of elements results in exponential reduction of the numerical error. The alignment of the grid with the direction of anisotropy is shown to further improve the accuracy of the solution. These effects are qualitatively explained and numerically quantified in 2- and 3-dimensional calculations with straight and curved anisotropy.     

The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate

We analyze the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities—the lower spectral radius and the largest Lyapunov exponent—are not algorithmically approximable.This work was completed while Blondel was visiting Tsitsiklis at MIT. This research was supported by the ARO under Grant DAAL-03-92-G-0115.     

Observability for two-dimensional systems

Sufficient conditions that a two-dimensional system with output is locally observable are presented. Known results depend on time derivatives of the output and the inverse function theorem. In some cases, no information is provided by these theories, and one must study observability by other methods. We dualize the observability problem to the controllability problem, and apply the deep results of Hermes on local controllability to prove a theorem concerning local observability.Research supported by NASA Ames Research Center under Grant NAG2-189 and the Joint Services Electronics Program under ONR Contract N0014-76-C1136.Research supported by NASA Ames Research Center under Grant NAG2-203 and the Joint Services Electronics Program under ONR Contract N0014-76-C1136.     

A Hybrid Fourier–Chebyshev Method for Partial Differential Equations

We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O(1/N) is sufficient for stability. R.B. Platte’s address after December 2009: Arizona State University, Department of Mathematics and Statistics, Tempe, AZ, 85287-1804.