Volume 5, Issue 2-4
Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations

Tao Tang & Xiang Xu

DOI:

Commun. Comput. Phys., 5 (2009), pp. 779-792.

Published online: 2009-02

Preview Full PDF 170 1770
Export citation
  • Abstract

It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods. More precisely, we can use spectral methods to postprocess numerical solutions of initial value differential equations. After a few number of iterations (say 3 to 4), the errors can decrease to a few orders of magnitude less. The iteration uses the Gauss-Seidel type strategy, which gives an explicit way of postprocessing. Numerical examples for ODEs, Hamiltonian system and integral equations are provided. They all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical solutions. In particular, for a Hamiltonian system accuracy is only one of the issues; some other conservative properties are even more important for large time simulations. The spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.

  • Keywords

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-5-779, author = {}, title = {Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations}, journal = {Communications in Computational Physics}, year = {2009}, volume = {5}, number = {2-4}, pages = {779--792}, abstract = {

It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods. More precisely, we can use spectral methods to postprocess numerical solutions of initial value differential equations. After a few number of iterations (say 3 to 4), the errors can decrease to a few orders of magnitude less. The iteration uses the Gauss-Seidel type strategy, which gives an explicit way of postprocessing. Numerical examples for ODEs, Hamiltonian system and integral equations are provided. They all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical solutions. In particular, for a Hamiltonian system accuracy is only one of the issues; some other conservative properties are even more important for large time simulations. The spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7763.html} }
TY - JOUR T1 - Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations JO - Communications in Computational Physics VL - 2-4 SP - 779 EP - 792 PY - 2009 DA - 2009/02 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7763.html KW - AB -

It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods. More precisely, we can use spectral methods to postprocess numerical solutions of initial value differential equations. After a few number of iterations (say 3 to 4), the errors can decrease to a few orders of magnitude less. The iteration uses the Gauss-Seidel type strategy, which gives an explicit way of postprocessing. Numerical examples for ODEs, Hamiltonian system and integral equations are provided. They all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical solutions. In particular, for a Hamiltonian system accuracy is only one of the issues; some other conservative properties are even more important for large time simulations. The spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.

Tao Tang & Xiang Xu. (2020). Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations. Communications in Computational Physics. 5 (2-4). 779-792. doi:
Copy to clipboard
The citation has been copied to your clipboard