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Article information
2020 , Volume 25, № 2, p.63-79
Idimeshev S.V.
Rational approximation for initial boundary problems with the fronts
A spectral method with adaptive rational approximation is proposed. In traditional spectral polynomial interpolation, the interpolation points are fixed, usually at the roots or extrema of orthogonal polynomials. Free selection of interpolation points is impossible due to the effect described in the Runge example. The key feature of rational interpolation is the free distribution of interpolation nodes without the occurrence of the Runge phenomenon. Nevertheless, in practice it is very important to implement rational approximation effectively. Here rational approximation is implemented using the barycentric Lagrange form. This leads to fast computations and numerical stability comparable with the polynomial interpolation. It is shown that rational interpolation has significant advantages over polynomial on functions that have singularities in the form of fronts. The key idea is that rational interpolation allows adapting interpolation points according to function singularities. An effective method of grid adaptation that accounts for singularity location was used. Method was generalized to the case of several singularities, for example, for solutions with several fronts. For the solutions of the Burgers equation with singularities in the form of fronts, it is shown that rational interpolation has significant advantages over polynomial. The implementation of spectral method is described, and calculations results on model problems, including problems with two fronts, are presented.
[full text]
Keywords: rational interpolation, polynomial interpolation, spectral method, singularity in the complex plane, barycentric Lagrange interpolation form, Burgers equation
doi: 10.25743/ICT.2020.25.2.006
Author(s):
Idimeshev Semyon Vasilyevich
Position: Junior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Akademika Rzhanova ave., 6
Phone Office: (383)330-93-61
E-mail: idimeshev@gmail.com
SPIN-code: 3793-6120 References:
[1] Boyd J.P. Chebyshev and Fourier spectral methods: second revised edition. New York: Dover Publications; 2000: 611.
[2] Trefethen L.N. Approximation theory and approximation practice. Society for Industrial and Applied Mathematics; 2013: 295.
[3] Higham N.J. The numerical stability of barycentric Lagrange interpolation. IMA Journal of Numerical Analysis. 2004; 24(4):547–556.
[4] Мысовских И.П. Лекции по методам вычислений. Санкт-Петербург: Изд-во Санкт-Петербургского ун-та; 1998: 784.
[5] Cheney W., Light W. Course in approximation theory. W. Light Brooks, Cole; 2000: 360.
[6] Baltensperger R., Berrut J.-P., Noel B. Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Mathematics of Computation. 1999; 68(227):1109–1121.
[7] Kosloff D., Tal-Ezer H. A modified Chebyshev pseudospectral method with an O(N-1) time step restriction. Journal of Computational Physics. 1993; 104(2):457–469.
[8] Jafari-Varzaneh H.A., Hosseini S.M. A new map for the Chebyshev pseudospectral solution of differential equations with large gradients. Numerical Algorithms. 2015; 69(1):95–108.
[9] Tee T.W., Trefethen L.N. A rational spectral collocation method with adaptively transformed chebyshev grid point. SIAM Journal on Scientific Computing. 2006; 28(5):1798–1811.
[10] Schneider C., Werner W. Some new aspects of rational interpolation. Mathematics of Computation. 1986; 47(175):285–299.
[11] Trefethen L.N. Spectral Methods in Matlab. Philadelphia: SIAM University City Center; 2000: 165.
[12] Холл Д., Уатт Д. Современные численные методы решения обыкновенных дифференциальных уравнений. М.: Мир; 1979: 312.
[13] Dormand J.R., Prince P.J. A family of embedded Runge—Kutta formulae. Journal of Computational and Applied Mathematics. 1980; 6(1):19–26.
[14] Shampine L.F., Reichelt M.W. The MATLAB ODE suite. SIAM Journal on Scientific Computing. 1997; 18(1):1–22.
[15] Huang W., Russell R.D. Adaptive moving mesh methods. AMS Book Sciences, vol. 174. New York: Springer; 2011: 432. DOI:10.1007/978-1-4419-7916-2
[16] Shapeev V.P., Vorozhtsov E.V. CAS application to the construction of the collocations and least residuals method for the solution of 3D Navier—Stokes equations. Lecture Notes in Computer Science. Computer Algebra in Scientific Computing. 2013; (8136):381–392.
Bibliography link:
Idimeshev S.V. Rational approximation for initial boundary problems with the fronts // Computational technologies. 2020. V. 25. № 2. P. 63-79
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