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Single and multi‐interval Legendre spectral methods in time for parabolic equations
Author(s) -
Tang Jianguo,
Ma Heping
Publication year - 2006
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20135
Subject(s) - mathematics , legendre polynomials , interval (graph theory) , dirichlet boundary condition , spectral method , parabolic partial differential equation , partial differential equation , algebraic equation , boundary value problem , mathematical analysis , neumann boundary condition , nonlinear system , dirichlet distribution , physics , combinatorics , quantum mechanics
In this article, we take the parabolic equation with Dirichlet boundary conditions as a model to present the Legendre spectral methods both in spatial and in time. Error analysis for the single/multi‐interval schemes in time is given. For the single interval spectral method in time, we obtain a better error estimate in L 2 ‐norm. For the multi‐interval spectral method in time, we obtain the L 2 ‐optimal error estimate in spatial. By choosing approximate trial and test functions, the methods result in algebraic systems with sparse forms. A parallel algorithm is constructed for the multi‐interval scheme in time. Numerical results show the efficiency of the methods. The methods are also applied to parabolic equations with Neumann boundary conditions, Robin boundary conditions and some nonlinear PDEs. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006