Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations

Abstract

We propose new Legendre–Gauss collocation algorithms for ordinary differential equations. We also design Legendre–Gauss-type collocation algorithms for time-dependent nonlinear partial differential equations. The suggested algorithms enjoy spectral accuracy in both time and space, and can be implemented in a fast and stable manner. Numerical results exhibit the effectiveness.

Keywords:

• spectral collocation method
• time-dependent nonlinear partial differential equations
• ordinary differential equations
• discretization in time and space
• high-order accuracy

1991 AMS Subject Classifications::

• 65M70
• 65L60
• 65L05
• 41A10
• 35K20
• 35L20

Acknowledgements

The work is supported in part by the NSF of China (No. 11171225, No. 11226330 and No. 11301343), the Innovation Program of Shanghai Municipal Education Commission (No. 12ZZ131), the Research Fund for the Doctoral Program of Higher Education of China (No. 20113127120002), the Research Fund for Young Teachers Program in Shanghai (No. shsf018), and the Fund for E-institute of Shanghai Universities (No. E03004).