Journal articles: 'Spectral collocation'

1

Hussaini, M. Y., D. A. Kopriva, and A. T. Patera. "Spectral collocation methods." Applied Numerical Mathematics 5, no. 3 (May 1989): 177–208. http://dx.doi.org/10.1016/0168-9274(89)90033-0.

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Zayernouri, Mohsen, and George Em Karniadakis. "Fractional Spectral Collocation Method." SIAM Journal on Scientific Computing 36, no. 1 (January 2014): A40—A62. http://dx.doi.org/10.1137/130933216.

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3

Li, Yiqun, Boying Wu, and Melvin Leok. "Spectral-collocation variational integrators." Journal of Computational Physics 332 (March 2017): 83–98. http://dx.doi.org/10.1016/j.jcp.2016.12.007.

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4

Heinrichs, Wilhelm. "Spectral Collocation on Triangular Elements." Journal of Computational Physics 145, no. 2 (September 1998): 743–57. http://dx.doi.org/10.1006/jcph.1998.6052.

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5

Gu, Zhendong, and Yanping Chen. "Piecewise Legendre spectral-collocation method for Volterra integro-differential equations." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 231–49. http://dx.doi.org/10.1112/s1461157014000485.

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Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in$h^{m}N^{-m}$-version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.

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6

Chantawansri, Tanya L., Su-Mi Hur, Carlos J. García-Cervera, Hector D. Ceniceros, and Glenn H. Fredrickson. "Spectral collocation methods for polymer brushes." Journal of Chemical Physics 134, no. 24 (June 28, 2011): 244905. http://dx.doi.org/10.1063/1.3604814.

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7

Hessari, Peyman, Sang Dong Kim, and Byeong-Chun Shin. "Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/780769.

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The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.

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8

WELLS, J. C., V. E. OBERACKER, M. R. STRAYER, and A. S. UMAR. "SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD." International Journal of Modern Physics C 06, no. 01 (February 1995): 143–67. http://dx.doi.org/10.1142/s0129183195000125.

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We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

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9

Ma, Xian, Yongxian Wang, Xiaoqian Zhu, Wei Liu, Wenbin Xiao, and Qiang Lan. "A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations." Entropy 23, no. 9 (September 18, 2021): 1227. http://dx.doi.org/10.3390/e23091227.

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The accuracy and efficiency of sound field calculations highly concern issues of hydroacoustics. Recently, one-dimensional spectral methods have shown high-precision characteristics when solving the sound field but can solve only simplified models of underwater acoustic propagation, thus their application range is small. Therefore, it is necessary to directly calculate the two-dimensional Helmholtz equation of ocean acoustic propagation. Here, we use the Chebyshev–Galerkin and Chebyshev collocation methods to solve the two-dimensional Helmholtz model equation. Then, the Chebyshev collocation method is used to model ocean acoustic propagation because, unlike the Galerkin method, the collocation method does not need stringent boundary conditions. Compared with the mature Kraken program, the Chebyshev collocation method exhibits a higher numerical accuracy. However, the shortcoming of the collocation method is that the computational efficiency cannot satisfy the requirements of real-time applications due to the large number of calculations. Then, we implemented the parallel code of the collocation method, which could effectively improve calculation effectiveness.

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Ma, Xian, Yongxian Wang, Xiaoqian Zhu, Wei Liu, Qiang Lan, and Wenbin Xiao. "A Spectral Method for Two-Dimensional Ocean Acoustic Propagation." Journal of Marine Science and Engineering 9, no. 8 (August 19, 2021): 892. http://dx.doi.org/10.3390/jmse9080892.

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The accurate calculation of the sound field is one of the most concerning issues in hydroacoustics. The one-dimensional spectral method has been used to correctly solve simplified underwater acoustic propagation models, but it is difficult to solve actual ocean acoustic fields using this model due to its application conditions and approximation error. Therefore, it is necessary to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation without using simplified models. Here, two commonly used spectral methods, Chebyshev–Galerkin and Chebyshev–collocation, are used to correctly solve the two-dimensional Helmholtz model equation. Since Chebyshev–collocation does not require harsh boundary conditions for the equation, it is then used to solve ocean acoustic propagation. The numerical calculation results are compared with analytical solutions to verify the correctness of the method. Compared with the mature Kraken program, the Chebyshev–collocation method exhibits higher numerical calculation accuracy. Therefore, the Chebyshev–collocation method can be used to directly solve the representative two-dimensional ocean acoustic propagation equation. Because there are no model constraints, the Chebyshev–collocation method has a wide range of applications and provides results with high accuracy, which is of great significance in the calculation of realistic ocean sound fields.

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11

Gheorghiu, C. I., M. E. Hochstenbach, B. Plestenjak, and J. Rommes. "Spectral collocation solutions to multiparameter Mathieu’s system." Applied Mathematics and Computation 218, no. 24 (August 2012): 11990–2000. http://dx.doi.org/10.1016/j.amc.2012.05.068.

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12

Benacchio, T., and L. Bonaventura. "Absorbing boundary conditions: a spectral collocation approach." International Journal for Numerical Methods in Fluids 72, no. 9 (January 4, 2013): 913–36. http://dx.doi.org/10.1002/fld.3768.

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13

Bialecki, B., and J. de Frutos. "ADI spectral collocation methods for parabolic problems." Journal of Computational Physics 229, no. 13 (July 2010): 5182–93. http://dx.doi.org/10.1016/j.jcp.2010.03.033.

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14

Zhang, Zhimin. "Superconvergence of a Chebyshev Spectral Collocation Method." Journal of Scientific Computing 34, no. 3 (October 16, 2007): 237–46. http://dx.doi.org/10.1007/s10915-007-9163-7.

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15

Heinrichs, Wilhelm. "Spectral collocation schemes on the unit disc." Journal of Computational Physics 199, no. 1 (September 2004): 66–86. http://dx.doi.org/10.1016/j.jcp.2004.02.001.

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16

Eisen, Henner, Wilhelm Heinrichs, and Kristian Witsch. "Spectral collocation methods and polar coordinate singularities." Journal of Computational Physics 91, no. 1 (November 1990): 251. http://dx.doi.org/10.1016/0021-9991(90)90022-s.

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17

Eisen, Henner, Wilhelm Heinrichs, and Kristian Witsch. "Spectral collocation methods and polar coordinate singularities." Journal of Computational Physics 96, no. 2 (October 1991): 241–57. http://dx.doi.org/10.1016/0021-9991(91)90235-d.

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18

Abdelkawy, M. A. "A Collocation Method Based on Jacobi and Fractional Order Jacobi Basis Functions for Multi-Dimensional Distributed-Order Diffusion Equations." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 7-8 (December 19, 2018): 781–92. http://dx.doi.org/10.1515/ijnsns-2018-0111.

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AbstractIn this work, shifted fractional-order Jacobi orthogonal function in the interval $[0,\mathcal{T}]$ is outputted of the classical Jacobi polynomial (see Definition 2.3). Also, we list and derive some facts related to the shifted fractional-order Jacobi orthogonal function. Spectral collocation techniques are addressed to solve the multidimensional distributed-order diffusion equations (MDODEs). A mixed of shifted Jacobi polynomials and shifted fractional order Jacobi orthogonal functions are used as basis functions to adapt the spatial and temporal discretizations, respectively. Based on the selected basis, a spectral collocation method is listed to approximate the MDODEs. By means of the selected basis functions, the given conditions are automatically satisfied. We conclude with the application of spectral collocation method for multi-dimensional distributed-order diffusion equations.

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19

Zaky, Mahmoud A., Eid H. Doha, Taha M. Taha, and Dumitru Baleanu. "NEW RECURSIVE APPROXIMATIONS FOR VARIABLE-ORDER FRACTIONAL OPERATORS WITH APPLICATIONS." Mathematical Modelling and Analysis 23, no. 2 (April 18, 2018): 227–39. http://dx.doi.org/10.3846/mma.2018.015.

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To broaden the range of applicability of variable-order fractional diﬀerential models, reliable numerical approaches are needed to solve the model equation.In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Speciﬁcally, we derive three-term recurrence relations to eﬃciently calculate the variable-order fractional integrals and derivatives of the modiﬁed generalized Laguerre polynomials, which lead to the corresponding fractional diﬀerentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.

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20

Muzara, Hillary, Stanford Shateyi, and Gerald Tendayi Marewo. "On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem." Open Physics 16, no. 1 (August 20, 2018): 554–62. http://dx.doi.org/10.1515/phys-2018-0072.

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AbstractIn this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

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21

Yang, Yin, and Yunqing Huang. "Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations." Advances in Mathematical Physics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/821327.

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We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

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22

Kim, Chang Ho, and U. Jin Choi. "Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, no. 3 (January 1998): 408–30. http://dx.doi.org/10.1017/s0334270000009474.

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AbstractWe propose and analyze the spectral collocation approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.

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23

Rashid, Abdur. "The Pseudo-Spectral Collocation Method for Resonant Long-Short Nonlinear Wave Interaction." Georgian Mathematical Journal 13, no. 1 (March 1, 2006): 143–52. http://dx.doi.org/10.1515/gmj.2006.143.

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Abstract A pseudo-spectral collocation method for a class of equations describing resonant long-short wave interaction is studied. Semi-discrete and fully discrete Fourier pseudo-spectral collocation schemes are given. In fully discrete case we establish a three-level explicit scheme which is convenient and saves time in real computation. We use energy estimation methods to obtain error estimates for the approximate solutions.

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24

JUNG, JAE-HUN, GAURAV KHANNA, and IAN NAGLE. "A SPECTRAL COLLOCATION APPROXIMATION FOR THE RADIAL-INFALL OF A COMPACT OBJECT INTO A SCHWARZSCHILD BLACK HOLE." International Journal of Modern Physics C 20, no. 11 (November 2009): 1827–48. http://dx.doi.org/10.1142/s012918310901476x.

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The inhomogeneous Zerilli equation is solved in time-domain numerically with the Chebyshev spectral collocation method to investigate a radial-infall of the point particle towards a Schwarzschild black hole. Singular source terms due to the point particle appear in the equation in the form of the Dirac δ-function and its derivative. For the approximation of singular source terms, we use the direct derivative projection method proposed in Ref. 9 without any regularization. The gravitational waveforms are evaluated as a function of time. We compare the results of the spectral collocation method with those of the explicit second-order central-difference method. The numerical results show that the spectral collocation approximation with the direct projection method is accurate and converges rapidly when compared with the finite-difference method.

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25

Donatelli, Marco, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers. "Spectral analysis and spectral symbol of matrices in isogeometric collocation methods." Mathematics of Computation 85, no. 300 (October 15, 2015): 1639–80. http://dx.doi.org/10.1090/mcom/3027.

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26

Abd-Elkawy, Mohamed A., and Rubayyi T. Alqahtani. "SPACE-TIME SPECTRAL COLLOCATION ALGORITHM FOR THE VARIABLE-ORDER GALILEI INVARIANT ADVECTION DIFFUSION EQUATIONS WITH A NONLINEAR SOURCE TERM." Mathematical Modelling and Analysis 22, no. 1 (January 11, 2017): 1–20. http://dx.doi.org/10.3846/13926292.2017.1258014.

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This paper presents a space-time spectral collocation technique for solving the variable-order Galilei invariant advection diffusion equation with a nonlinear source term (VO-NGIADE). We develop a collocation scheme to approximate VONGIADE by means of the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) and shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods. We successfully extend the proposed technique to solve the two-dimensional space VO-NGIADE. The discussed numerical tests illustrate the capability and high accuracy of the proposed methodologies.

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27

Youssri, Y. H., and R. M. Hafez. "Exponential Jacobi spectral method for hyperbolic partial differential equations." Mathematical Sciences 13, no. 4 (September 26, 2019): 347–54. http://dx.doi.org/10.1007/s40096-019-00304-w.

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Abstract Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments.

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Zogheib, Bashar, Emran Tohidi, and Stanford Shateyi. "Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions." Advances in Mathematical Physics 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/5691452.

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A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.

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Jackiewicz, Z., and B. Zubik-Kowal. "Spectral Collocation and Waveform Relaxation Methods with Gegenbauer Geconstruction for Nonlinear Conservation Laws." Computational Methods in Applied Mathematics 5, no. 1 (2005): 51–71. http://dx.doi.org/10.2478/cmam-2005-0002.

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Abstract We investigate the Chebyshev spectral collocation and waveform relaxation methods for nonlinear conservation laws. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated by highly stable implicit methods. The obtained numerical solution is then enhanced on the intervals of smoothness by the Gegenbauer reconstruction. The effectiveness of this approach is illustrated by numerical experiments.

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30

Abdelkawy, Mohamed A., Zulqurnain Sabir, Juan L. G. Guirao, and Tareq Saeed. "Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model." Open Physics 18, no. 1 (November 20, 2020): 770–78. http://dx.doi.org/10.1515/phys-2020-0185.

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AbstractThe present study aims to design a second-order nonlinear Lane–Emden coupled functional differential model and numerically investigate by using the famous spectral collocation method. For validation of the newly designed model, three dissimilar variants have been considered and formulated numerically by applying a famous spectral collocation method. Moreover, a comparison of the obtained results with the exact/true results endorses the effectiveness and competency of the newly designed model, as well as the present technique.

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31

Mkhatshwa, Musawenkhosi P., Sandile S. Motsa, and Precious Sibanda. "Numerical solution of time-dependent Emden-Fowler equations using bivariate spectral collocation method on overlapping grids." Nonlinear Engineering 9, no. 1 (June 19, 2020): 299–318. http://dx.doi.org/10.1515/nleng-2020-0017.

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AbstractIn this work, we present a new modification to the bivariate spectral collocation method in solving Emden-Fowler equations. The novelty of the modified approach is the use of overlapping grids when applying the Chebyshev spectral collocation method. In the case of nonlinear partial differential equations, the quasilinearisation method is used to linearize the equation. The multi-domain technique is applied in both space and time intervals, which are both decomposed into overlapping subintervals. The spectral collocation method is then employed in the discretization of the iterative scheme to give a matrix system to be solved simultaneously across the overlapping subintervals. Several test examples are considered to demonstrate the general performance of the numerical technique in terms of efficiency and accuracy. The numerical solutions are matched against exact solutions to confirm the accuracy and convergence of the method. The error bound theorems and proofs have been considered to emphasize on the benefits of the method. The use of an overlapping grid gives a matrix system with less dense matrices that can be inverted in a computationally efficient manner. Thus, implementing the spectral collocation method on overlapping grids improves the computational time and accuracy. Furthermore, few grid points in each subinterval are required to achieve stable and accurate results. The approximate solutions are established to be in excellent agreement with the exact analytical solutions.

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32

Huang, Can, and Zhimin Zhang. "The spectral collocation method for stochastic differential equations." Discrete & Continuous Dynamical Systems - B 18, no. 3 (2013): 667–79. http://dx.doi.org/10.3934/dcdsb.2013.18.667.

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33

Cividini, A., and E. Zampieri. "Nonlinear stress analysis problems by spectral collocation methods." Computer Methods in Applied Mechanics and Engineering 145, no. 1-2 (June 1997): 185–201. http://dx.doi.org/10.1016/s0045-7825(96)01195-4.

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34

Parter, Seymour V., and Ernest E. Rothman. "Preconditioning Legendre Spectral Collocation Approximations to Elliptic Problems." SIAM Journal on Numerical Analysis 32, no. 2 (April 1995): 333–85. http://dx.doi.org/10.1137/0732015.

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35

Kim, Sang Dong, and Seymour V. Parter. "Preconditioning Chebyshev Spectral Collocation by Finite-Difference Operators." SIAM Journal on Numerical Analysis 34, no. 3 (June 1997): 939–58. http://dx.doi.org/10.1137/s0036142995285034.

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36

Sun, Weiwei. "Spectral Analysis of Hermite Cubic Spline Collocation Systems." SIAM Journal on Numerical Analysis 36, no. 6 (January 1999): 1962–75. http://dx.doi.org/10.1137/s0036142997322722.

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McCoid, Conor, and Manfred R. Trummer. "Improved Resolution of Boundary Layers for Spectral Collocation." SIAM Journal on Scientific Computing 41, no. 5 (January 2019): A2836—A2849. http://dx.doi.org/10.1137/18m1211015.

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38

Quarteroni, Alfio, and Giovanni Sacchi-Landriani. "Domain decomposition preconditioners for the spectral collocation method." Journal of Scientific Computing 3, no. 1 (March 1988): 45–76. http://dx.doi.org/10.1007/bf01066482.

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39

Pozrikidis, C. "A spectral collocation method with triangular boundary elements." Engineering Analysis with Boundary Elements 30, no. 4 (April 2006): 315–24. http://dx.doi.org/10.1016/j.enganabound.2005.11.005.

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40

Du, Kui. "On Well-Conditioned Spectral Collocation and Spectral Methods by the Integral Reformulation." SIAM Journal on Scientific Computing 38, no. 5 (January 2016): A3247—A3263. http://dx.doi.org/10.1137/15m1046629.

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Li, Yiqun, Boying Wu, and Melvin Leok. "Construction and comparison of multidimensional spectral variational integrators and spectral collocation methods." Applied Numerical Mathematics 132 (October 2018): 35–50. http://dx.doi.org/10.1016/j.apnum.2018.05.010.

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42

Cacuci, Dan Gabriel. "First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: II. Application to a Nuclear Reactor Heat Removal Benchmark." Journal of Nuclear Engineering 1, no. 1 (September 9, 2020): 18–45. http://dx.doi.org/10.3390/jne1010003.

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This work illustrates the application of a comprehensive first-order adjoint sensitivity analysis methodology (1st-CASAM) to a heat conduction and convection analytical benchmark problem which simulates heat removal from a nuclear reactor fuel rod. This analytical benchmark problem can be used to verify the accuracy of numerical solutions provided by software modeling heat transport and fluid flow systems. This illustrative heat transport benchmark shows that collocation methods require one adjoint computation for every collocation point while spectral expansion methods require one adjoint computation for each cardinal function appearing in the respective expansion when recursion relations cannot be developed between the corresponding adjoint functions. However, it is also shown that spectral methods are much more efficient when recursion relations provided by orthogonal polynomials make it possible to develop recursion relations for computing the corresponding adjoint functions. When recursion relations cannot be developed for the adjoint functions, the collocation method is probably more efficient than the spectral expansion method, since the sources for the corresponding adjoint systems are just Dirac delta functions (which makes the respective computation equivalent to the computation of a Green’s function), rather than the more elaborated sources involving high-order Fourier basis functions or orthogonal polynomials. For systems involving many independent variables, it is likely that a hybrid combination of spectral expansions in some independent variables and collocation in the remaining independent variables would provide the most efficient computational outcome.

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Gheorghiu, Călin-Ioan. "Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems." Symmetry 13, no. 3 (February 27, 2021): 385. http://dx.doi.org/10.3390/sym13030385.

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This work is about the use of some classical spectral collocation methods as well as with the new software system Chebfun in order to compute the eigenpairs of some high order Sturm–Liouville eigenproblems. The analysis is divided into two distinct directions. For problems with clamped boundary conditions, we use the preconditioning of the spectral collocation differentiation matrices and for hinged end boundary conditions the equation is transformed into a second order system and then the conventional ChC is applied. A challenging set of “hard” benchmark problems, for which usual numerical methods (FD, FE, shooting, etc.) encounter difficulties or even fail, are analyzed in order to evaluate the qualities and drawbacks of spectral methods. In order to separate “good” and “bad” (spurious) eigenvalues, we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation N. This drift gives us a very precise indication of the accuracy with which the eigenvalues are computed, i.e., an automatic estimation and error control of the eigenvalue error. Two MATLAB codes models for spectral collocation (ChC and SiC) and another for Chebfun are provided. They outperform the old codes used so far and can be easily modified to solve other problems.

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Abdelkawy, Mohamed A., Ahmed Z. M. Amin, Mohammed M. Babatin, Abeer S. Alnahdi, Mahmoud A. Zaky, and Ramy M. Hafez. "Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations." Fractal and Fractional 5, no. 3 (September 9, 2021): 115. http://dx.doi.org/10.3390/fractalfract5030115.

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In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme.

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45

GHOREISHI, F., and P. MOKHTARY. "SPECTRAL COLLOCATION METHOD FOR MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATIONS." International Journal of Computational Methods 11, no. 05 (October 2014): 1350072. http://dx.doi.org/10.1142/s0219876213500722.

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In this paper, the spectral collocation method is investigated for the numerical solution of multi-order Fractional Differential Equations (FDEs). We choose the orthogonal Jacobi polynomials and set of Jacobi Gauss–Lobatto quadrature points as basis functions and grid points respectively. This solution strategy is an application of the matrix-vector-product approach in spectral approximation of FDEs. The fractional derivatives are described in the Caputo type. Numerical solvability and an efficient convergence analysis of the method have also been discussed. Due to the fact that the solutions of fractional differential equations usually have a weak singularity at origin, we use a variable transformation method to change some classes of the original equation into a new equation with a unique smooth solution such that, the spectral collocation method can be applied conveniently. We prove that after this regularization technique, numerical solution of the new equation has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method.

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46

Abd-Elhameed, W. M., E. H. Doha, and Y. H. Youssri. "New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/542839.

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This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions.

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SEO, JEONG-KWEON, and BYEONG-CHUN SHIN. "LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS." Honam Mathematical Journal 37, no. 3 (September 25, 2015): 299–315. http://dx.doi.org/10.5831/hmj.2015.37.3.299.

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Denolle, M. A., E. M. Dunham, and G. C. Beroza. "Solving the Surface-Wave Eigenproblem with Chebyshev Spectral Collocation." Bulletin of the Seismological Society of America 102, no. 3 (June 1, 2012): 1214–23. http://dx.doi.org/10.1785/0120110183.

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Sun, Ya-Song, and Ben-Wen Li. "Spectral Collocation Method for Transient Conduction-Radiation Heat Transfer." Journal of Thermophysics and Heat Transfer 24, no. 4 (October 2010): 823–32. http://dx.doi.org/10.2514/1.43400.

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Gu, Dong-qin. "ADI-Spectral Collocation Methods for Two-Dimensional Parabolic Equations." East Asian Journal on Applied Mathematics 10, no. 2 (June 2020): 399–419. http://dx.doi.org/10.4208/eajam.300819.271019.

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Journal articles on the topic 'Spectral collocation'

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