Volume 47, Issue 2
Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems

Fei Liu

J. Math. Study, 47 (2014), pp. 190-207.

Published online: 2014-06

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  • Abstract

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.

  • Keywords

Spectral-tau method, variable coefficients, separable equation, Thomas-Fermi equation.

  • AMS Subject Headings

65M70, 65N35, 65L10, 33C45

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liufei84@gmail.com (Fei Liu)

  • BibTex
  • RIS
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@Article{JMS-47-190, author = {Liu , Fei}, title = {Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {2}, pages = {190--207}, abstract = {

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n2.14.04}, url = {http://global-sci.org/intro/article_detail/jms/9954.html} }
TY - JOUR T1 - Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems AU - Liu , Fei JO - Journal of Mathematical Study VL - 2 SP - 190 EP - 207 PY - 2014 DA - 2014/06 SN - 47 DO - http://doi.org/10.4208/jms.v47n2.14.04 UR - https://global-sci.org/intro/article_detail/jms/9954.html KW - Spectral-tau method, variable coefficients, separable equation, Thomas-Fermi equation. AB -

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.

Fei Liu. (2019). Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems. Journal of Mathematical Study. 47 (2). 190-207. doi:10.4208/jms.v47n2.14.04
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