Abstract
Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.
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References
Guo, B. Y. Spectral Methods and Their Applications, World Scientific, Singapore (1998)
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. Spectral Methods: Fundamentals in Single Domains (Scientific Computation), Springer-Verlag, Berlin (2006)
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation), Springer-Verlag, Berlin (2007)
Pavoni, D. Single and multidomain Chebyshev collocation methods for the Korteweg-de Vries equation. Calcolo, 25, 311–346 (1988)
Quarteroni, A. Domain decomposition methods for systems of conservation laws: spectral collocation approximations. SIAM J. Sci. Stat. Comput., 11, 1029–1052 (1990)
Funaro, D. Domain decomposition methods for pseudospectral approximations, part I, second order equations in one dimension. Numer. Math., 52, 329–344 (1988)
Heinrichs, W. Domain decomposition for fourth-order problems. SIAM J. Numer. Anal., 30, 435–453 (1993)
Gervasio, P. and Saleri, F. Stabilized spectral element approximation for the Navier-Stokes equations. Numer. Meth. Part. D. E., 14, 115–141 (1998)
Szabó, B. and Babuška, I. Finite Element Analysis, John Wiley & Sons, New York (1991)
Shen, J. Efficient spectral-Galerkin method I, direct solvers for second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput., 15, 1489–1505 (1994)
Karniadakis, G. E. and Sherwin, S. J. Spectral/hp Element Methods for CFD, Numerical Mathematics and Scientific Computation, Oxford University Press, New York (1999)
Bernardi, C. and Maday, Y. Polynomial interpolation results in Sobolev space. J. Comput. Appl. Math., 43, 53–82 (1992)
Ciarlet, P. G. The Finite Element Method for Elliptic Problems, North Holland, Amsterdam (1978)
Schwab, C. p- and hp-Finite Element Methods, Numerical Mathematics and Scientific Computation, Oxford University Press, New York (1998)
Ma, H. P. and Guo, B. Y. Composite Legendre-Laguerre pseudospectral approximation in unbounded domains. IMA J. Numer. Anal., 21, 587–602 (2001)
Guo, B. Y. and Ma, H. P. Composite Legendre-Laguerre approximation in unbounded domains. J. Comput. Math., 19, 101–112 (2001)
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Communicated by Shi-qiang DAI
Project supported by the National Natural Science Foundation of China (No. 60874039) and the Leading Academic Discipline Project of Shanghai Municipal Education Commission (No. J50101)
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Ji, Yy., Wu, H., Ma, Hp. et al. Multidomain pseudospectral methods for nonlinear convection-diffusion equations. Appl. Math. Mech.-Engl. Ed. 32, 1255–1268 (2011). https://doi.org/10.1007/s10483-011-1498-9
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DOI: https://doi.org/10.1007/s10483-011-1498-9